<?xml version="1.0"?>
<div class="add">

<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>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</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="E1:126"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:126"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T65" target="_self">Th65</a></i>;<br/>
<a NAME="E3:126"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:126"><i><font color="Green">E104</font></i></a>, <a class="ref" href="euclidlp.html#T71" target="_self">Th71</a></i>;<br/>
<b>assume </b><a NAME="E4:126"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</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="E6:126"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T54" target="_self">Th54</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</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="E8:126"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E4:126"><i><font color="Green">E107</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E9:126"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:126"><i><font color="Green">E104</font></i></a>, <a class="txt" href="euclidlp.html#E2:126"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclidlp.html#T76" target="_self">Th76</a></i>;<br/>
<a NAME="E10:126"/><b>then </b><i><font color="Green">E111</font></i>: 
( not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:126"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E8:126"><i><font color="Green">E109</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self">Th54</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:126"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:126"><i><font color="Green">E106</font></i></a>, <a class="txt" href="euclidlp.html#E10:126"><i><font color="Green">E111</font></i></a>, <a class="ref" href="euclidlp.html#T67" target="_self">Th67</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="E14:126"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E12:126"><i><font color="Green">E112</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E15:126"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E12:126"><i><font color="Green">E112</font></i></a>, <a class="txt" href="euclidlp.html#E1:126"><i><font color="Green">E104</font></i></a>, <a class="ref" href="euclidlp.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E16:126"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:126"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E9:126"><i><font color="Green">E110</font></i></a>, <a class="txt" href="euclidlp.html#E14:126"><i><font color="Green">E113</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E17:126"/><b>then </b><i><font color="Green">E116</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E12:126"><i><font color="Green">E112</font></i></a>, <a class="txt" href="euclidlp.html#E14:126"><i><font color="Green">E113</font></i></a>, <a class="ref" href="euclidlp.html#T69" target="_self">Th69</a></i>;<br/>
<a NAME="E18:126"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:126"><i><font color="Green">E104</font></i></a>, <a class="txt" href="euclidlp.html#E6:126"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E14:126"><i><font color="Green">E113</font></i></a>, <a class="txt" href="euclidlp.html#E17:126"><i><font color="Green">E116</font></i></a>, <a class="ref" href="euclidlp.html#T100" target="_self">Th100</a></i>;<br/>
<a NAME="E19:126"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="euclidlp.html#R7" target="_self">are_coplane</a> 
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:126"><i><font color="Green">E104</font></i></a>, <a class="ref" href="euclidlp.html#T101" target="_self">Th101</a></i>;<br/>
<a NAME="E20:126"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E15:126"><i><font color="Green">E114</font></i></a>, <a class="ref" href="euclidlp.html#T114" target="_self">Th114</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</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="E22:126"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T54" target="_self">Th54</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</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="E24:126"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:126"><i><font color="Green">E106</font></i></a>, <a class="txt" href="euclidlp.html#E22:126"><i><font color="Green">E118</font></i></a>, <a class="ref" href="euclidlp.html#T58" target="_self">Th58</a></i>;<br/>
<a NAME="E25:126"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E24:126"><i><font color="Green">E119</font></i></a>, <a class="txt" href="euclidlp.html#E22:126"><i><font color="Green">E118</font></i></a>, <a class="ref" href="euclidlp.html#T69" target="_self">Th69</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E27:126"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E12:126"><i><font color="Green">E112</font></i></a>, <a class="txt" href="euclidlp.html#E14:126"><i><font color="Green">E113</font></i></a>, <a class="txt" href="euclidlp.html#E22:126"><i><font color="Green">E118</font></i></a>, <a class="txt" href="euclidlp.html#E16:126"><i><font color="Green">E115</font></i></a>, <a class="ref" href="euclidlp.html#T75" target="_self">Th75</a></i>;<br/>
<a NAME="E28:126"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:126"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E8:126"><i><font color="Green">E109</font></i></a>, <a class="txt" href="euclidlp.html#E9:126"><i><font color="Green">E110</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E29:126"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:126"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E8:126"><i><font color="Green">E109</font></i></a>, <a class="ref" href="euclidlp.html#T69" target="_self">Th69</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="126_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:126_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="126_1_1_1"><b>case </b></a><a NAME="E1:126_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:126_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E17:126"><i><font color="Green">E116</font></i></a>, <a class="txt" href="euclidlp.html#E20:126"><i><font color="Green">E117</font></i></a>, <a class="txt" href="euclidlp.html#E6:126"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E8:126"><i><font color="Green">E109</font></i></a>, <a class="txt" href="euclidlp.html#E16:126"><i><font color="Green">E115</font></i></a>, <a class="ref" href="euclidlp.html#T69" target="_self">Th69</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="126_1_1_2"><b>case </b></a><a NAME="E1:126_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><a NAME="E2:126_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E8:126"><i><font color="Green">E109</font></i></a>, <a class="txt" href="euclidlp.html#E14:126"><i><font color="Green">E113</font></i></a>, <a class="txt" href="euclidlp.html#E24:126"><i><font color="Green">E119</font></i></a>, <a class="txt" href="euclidlp.html#E22:126"><i><font color="Green">E118</font></i></a>, <a class="txt" href="euclidlp.html#E1:126"><i><font color="Green">E104</font></i></a>, <a class="ref" href="euclidlp.html#T82" target="_self">Th82</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:126_1_1_2"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T37" target="_self">Th37</a></i>;<br/><i><font color="Green">E124</font></i>: <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font> = 
1 <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E4:126_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E4:126_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a></i>
;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></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>8</sub></font></span>)</span>;<br/><a NAME="E6:126_1_1_2"/><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></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>8</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E29:126"><i><font color="Green">E122</font></i></a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></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>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <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"><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> <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> <font color="Maroon">c<sub>8</sub></font></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>7</sub></font> <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> <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> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></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> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>7</sub></font>
<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>13</sub></font> <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> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self">Th17</a></i>
<br/>.= 
<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>8</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></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>
<br/>.= 
<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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <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></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
<br/>.= 
<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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <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></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="euclidlp.html#T7" target="_self">Th7</a></i>
<br/>.= 
<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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K7">+</a> <span class="p5">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <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></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>7</sub></font>
<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>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>7</sub></font>
<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>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><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>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>
<br/>.= 
<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"><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>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</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> <font color="Maroon">c<sub>7</sub></font>
<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>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</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>10</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> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="euclidlp.html#E5:126_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="euclid_4.html#T6">EUCLID_4:6</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>13</sub></font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</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>10</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> <font color="Maroon">c<sub>7</sub></font>
<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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</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>10</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> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</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>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a>, <a class="txt" href="euclidlp.html#E27:126"><i><font color="Green">E121</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</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>11</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> <font color="Maroon">c<sub>7</sub></font></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>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</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> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</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>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></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"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</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>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>12</sub></font></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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</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>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>12</sub></font></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">1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</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>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></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>13</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self">Th16</a></i>
;<br/><a NAME="E8:126_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></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>8</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E25:126"><i><font color="Green">E120</font></i></a></i>;<br/><b>hence </b><a NAME="E9:126_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E6:126_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self">Th54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E31:126"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


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