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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> 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>assume </b><a NAME="E1:112"/><i><font color="Green">E93</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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="euclidlp.html#R7" target="_self">are_coplane</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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="E3:112"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#D12" target="_self">Def12</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>assume </b><a NAME="E4:112"/><i><font color="Green">E95</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>
 ;<br/>

<a NAME="E5:112"/><i><font color="Green">E96</font></i>: 
 <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</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>consider </b><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="E7:112"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <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#E1:112"><i><font color="Green">E93</font></i></a>, <a class="ref" href="euclid_4.html#D2">EUCLID_4:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font>, <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="E9:112"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">c<sub>10</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>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:112"><i><font color="Green">E93</font></i></a>, <a class="ref" href="euclid_4.html#D2">EUCLID_4:def 2</a></i>;<br/>
<a NAME="E10:112"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</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#E7:112"><i><font color="Green">E97</font></i></a>, <a class="txt" href="euclidlp.html#E9:112"><i><font color="Green">E98</font></i></a>, <a class="ref" href="euclid_4.html#T10">EUCLID_4:10</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="E12:112"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="euclid_4.html#R1">are_orthogonal</a>  )
 <b>by </b><i><a class="ref" href="euclidlp.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E13:112"/><i><font color="Green">E101</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>10</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E7:112"><i><font color="Green">E97</font></i></a>, <a class="txt" href="euclidlp.html#E12:112"><i><font color="Green">E100</font></i></a>, <a class="txt" href="euclidlp.html#E10:112"><i><font color="Green">E99</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="txt" href="euclidlp.html#E4:112"><i><font color="Green">E95</font></i></a></i>;<br/>
<a NAME="E14:112"/><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>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E7:112"><i><font color="Green">E97</font></i></a>, <a class="txt" href="euclidlp.html#E13:112"><i><font color="Green">E101</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E15:112"/>
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E14:112"><i><font color="Green">E102</font></i></a>, <a class="txt" href="euclidlp.html#E12:112"><i><font color="Green">E100</font></i></a>, <a class="ref" href="euclidlp.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E16:112"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a> 
 
<b>by </b><i><a class="ref" href="euclidlp.html#T50" target="_self">Th50</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</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="E18:112"/><i><font color="Green">E104</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>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:112"><i><font color="Green">E93</font></i></a>, <a class="ref" href="euclid_4.html#T15">EUCLID_4:15</a></i>;<br/>
<a NAME="E19:112"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="ref" href="euclid_4.html#T10">EUCLID_4:10</a></i>;<br/>
<a NAME="E20:112"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a> 
 
<b>by </b><i><a class="txt" href="euclidlp.html#E7:112"><i><font color="Green">E97</font></i></a>, <a class="txt" href="euclidlp.html#E12:112"><i><font color="Green">E100</font></i></a>, <a class="txt" href="euclidlp.html#E13:112"><i><font color="Green">E101</font></i></a>, <a class="txt" href="euclidlp.html#E16:112"><i><font color="Green">E103</font></i></a>, <a class="txt" href="euclidlp.html#E18:112"><i><font color="Green">E104</font></i></a>, <a class="ref" href="euclidlp.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E21:112"/><i><font color="Green">E106</font></i>: 
 <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<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="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E7:112"><i><font color="Green">E97</font></i></a>, <a class="txt" href="euclidlp.html#E3:112"><i><font color="Green">E94</font></i></a>, <a class="txt" href="euclidlp.html#E12:112"><i><font color="Green">E100</font></i></a>, <a class="txt" href="euclidlp.html#E18:112"><i><font color="Green">E104</font></i></a>, <a class="txt" href="euclidlp.html#E5:112"><i><font color="Green">E96</font></i></a>, <a class="txt" href="euclidlp.html#E10:112"><i><font color="Green">E99</font></i></a>, <a class="txt" href="euclidlp.html#E20:112"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclidlp.html#T97" target="_self">Th97</a></i>;<br/>
<b>take </b>
 <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E22:112"/>
(  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> is    <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> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> is <a href="euclidlp.html#V1" target="_self">being_plane</a> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E5:112"><i><font color="Green">E96</font></i></a>, <a class="txt" href="euclidlp.html#E3:112"><i><font color="Green">E94</font></i></a>, <a class="txt" href="euclidlp.html#E21:112"><i><font color="Green">E106</font></i></a>, <a class="txt" href="euclidlp.html#E20:112"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclidlp.html#D10" target="_self">Def10</a></i>;<br/>


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