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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>









<b>assume </b><a NAME="E1:105"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a>  )
 ;<br/>

<a NAME="E2:105"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:105"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:105"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E4:105"><i><font color="Green">E89</font></i></a></i>;<br/>
<a NAME="E7:105"/><i><font color="Green">E91</font></i>: 
 <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:105"><i><font color="Green">E88</font></i></a>, <a class="txt" href="euclidlp.html#E6:105"><i><font color="Green">E90</font></i></a>, <a class="ref" href="euclidlp.html#T88" target="_self">Th88</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:105"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:105"><i><font color="Green">E88</font></i></a>, <a class="txt" href="euclidlp.html#E6:105"><i><font color="Green">E90</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:105"/><i><font color="Green">E93</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E9:105"><i><font color="Green">E92</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:105"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:105"><i><font color="Green">E88</font></i></a>, <a class="txt" href="euclidlp.html#E6:105"><i><font color="Green">E90</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E15:105"/><i><font color="Green">E95</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E13:105"><i><font color="Green">E94</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E17:105"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:105"><i><font color="Green">E88</font></i></a>, <a class="txt" href="euclidlp.html#E6:105"><i><font color="Green">E90</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E19:105"/><i><font color="Green">E97</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E17:105"><i><font color="Green">E96</font></i></a></i>;<br/>
<a NAME="E20:105"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E11:105"><i><font color="Green">E93</font></i></a>, <a class="ref" href="euclidlp.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E21:105"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E19:105"><i><font color="Green">E97</font></i></a>, <a class="ref" href="euclidlp.html#T32" target="_self">Th32</a></i>;<br/>
<i><font color="Green">E100</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E20:105"><i><font color="Green">E98</font></i></a>, <a class="txt" href="euclidlp.html#E15:105"><i><font color="Green">E95</font></i></a>, <a class="ref" href="euclidlp.html#T32" target="_self">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T22" target="_self">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>
<i><font color="Green">E101</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E11:105"><i><font color="Green">E93</font></i></a>, <a class="ref" href="euclidlp.html#T32" target="_self">Th32</a>, <a class="txt" href="euclidlp.html#E21:105"><i><font color="Green">E99</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T22" target="_self">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E25:105"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:105"><i><font color="Green">E88</font></i></a>, <a class="txt" href="euclidlp.html#E22:105"><i><font color="Green">E100</font></i></a>, <a class="txt" href="euclidlp.html#E23:105"><i><font color="Green">E101</font></i></a>, <a class="ref" href="euclidlp.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E26:105"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E20:105"><i><font color="Green">E98</font></i></a>, <a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:105_3"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:105_3"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="euclidlp.html#E6:105"><i><font color="Green">E90</font></i></a>, <a class="txt" href="euclidlp.html#E1:105_3"><i><font color="Green">E104</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</sub></font>, <font color="Maroon">c<sub>30</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:105_3"/><i><font color="Green">E106</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:105_3"><i><font color="Green">E105</font></i></a></i>;<br/><font color="Maroon">c<sub>26</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E5:105_3"><i><font color="Green">E106</font></i></a>, <a class="ref" href="euclidlp.html#T32" target="_self">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E26:105"><i><font color="Green">E103</font></i></a>, <a class="ref" href="euclidlp.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E25:105"><i><font color="Green">E102</font></i></a>, <a class="ref" href="euclid_4.html#T6">EUCLID_4:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T6">EUCLID_4:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</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">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</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">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</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">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T7">EUCLID_4:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T7">EUCLID_4:7</a></i>
;<br/><b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:105_3"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</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">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="euclidlp.html#T33" target="_self">Th33</a></i>;<br/><b>thus </b><a NAME="E9:105_3"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E8:105_3"><i><font color="Green">E107</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E28:105"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E29:105"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E7:105"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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