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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L0</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">n</font>;<br/><b>let </b><font color="Maroon">L1</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">n</font>;<br/><b>let </b><font color="Maroon">L2</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">n</font>;<br/><b>let </b><font color="Maroon">P</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">n</font>;<br/>











<b>assume </b><a NAME="E1:122"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">L0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L0</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 ;<br/>

<b>assume </b><a NAME="E2:122"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">L0</font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">L1</font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">L2</font> )
 ;<br/>

<a NAME="E3:122"/><b>then </b><i><font color="Green">E88</font></i>: 
( <font color="Maroon">L0</font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  &amp; <font color="Maroon">L1</font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  &amp; <font color="Maroon">L2</font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E15"><i><font color="Green">Lemma69</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">x0</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E5:122"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L0</font> &amp; not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, , <a class="ref" href="euclidlp.html#T25" target="_self">Th25</a></i>;<br/>
<b>consider </b><font color="Maroon">x2</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E7:122"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> &amp; not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, , <a class="ref" href="euclidlp.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E8:122"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">L0</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, , <a class="txt" href="euclidlp.html#E2:122"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E9:122"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, , , , <a class="ref" href="euclidlp.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E10:122"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font>,<font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E11:122"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x</font>,<font color="Maroon">x0</font>,<font color="Maroon">x2</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, , , <a class="txt" href="euclidlp.html#E10:122"><i><font color="Green">E107</font></i></a>, <a class="ref" href="euclidlp.html#T33" target="_self">Th33</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E13:122"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, , <a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E15:122"/><i><font color="Green">E110</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, <a class="txt" href="euclidlp.html#E13:122"><i><font color="Green">E109</font></i></a>, <a class="txt" href="euclidlp.html#E11:122"><i><font color="Green">E108</font></i></a>, <a class="ref" href="euclidlp.html#T30" target="_self">Th30</a></i>;<br/>
<i><font color="Green">E111</font></i>: <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x</font></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">a1</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="euclidlp.html#E15:122"><i><font color="Green">E110</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E15:122"><i><font color="Green">E110</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>
<a NAME="E17:122"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, <a class="txt" href="euclidlp.html#E13:122"><i><font color="Green">E109</font></i></a>, <a class="txt" href="euclidlp.html#E2:122"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E18:122"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, , <a class="txt" href="euclidlp.html#E13:122"><i><font color="Green">E109</font></i></a>, , <a class="ref" href="euclidlp.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E19:122"/>
<font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font>,<font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="euclid_4.html#R1">are_orthogonal</a> 
 
<b>by </b><i><a class="txt" href="euclidlp.html#E9:122"><i><font color="Green">E106</font></i></a>, </i>;<br/>
<a NAME="E20:122"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="euclid_4.html#D6">EUCLID_4:def 6</a></i>;<br/>
<a NAME="E21:122"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font>,<font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="euclid_4.html#R1">are_orthogonal</a> 
 
<b>by </b><i><a class="txt" href="euclidlp.html#E18:122"><i><font color="Green">E113</font></i></a>, </i>;<br/>
<a NAME="E22:122"/>
<font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E23:122"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="euclid_4.html#T22">EUCLID_4:22</a></i>;<br/>
0 = 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="txt" href="euclidlp.html#E16:122"><i><font color="Green">E111</font></i></a>, <a class="ref" href="euclidlp.html#T42" target="_self">Th42</a>, <a class="ref" href="euclid_4.html#D6">EUCLID_4:def 6</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T25">EUCLID_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T26">EUCLID_4:26</a></i>
<br/>.= 
<font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="txt" href="euclidlp.html#E20:122"><i><font color="Green">E114</font></i></a>, <a class="ref" href="euclid_4.html#T26">EUCLID_4:26</a></i>
;<br/>
<b>then </b><i><font color="Green">E153</font></i>: <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E16:122"><i><font color="Green">E111</font></i></a>, <a class="ref" href="euclidlp.html#T45" target="_self">Th45</a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>
<a NAME="E26:122"/>
( <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E13:122"><i><font color="Green">E109</font></i></a>, , </i>;<br/>
<a NAME="E27:122"/><b>then </b>
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E51"><i><font color="Green">Lemma117</font></i></a>, </i>;<br/>
<a NAME="E28:122"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">L1</font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">L0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E17:122"><i><font color="Green">E112</font></i></a>, <a class="txt" href="euclidlp.html#E8:122"><i><font color="Green">E105</font></i></a>, <a class="txt" href="euclidlp.html#E8"><i><font color="Green">Lemma60</font></i></a></i>;<br/>
<a NAME="E29:122"/>
<font color="Maroon">L1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">L0</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, <a class="ref" href="euclidlp.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E30:122"/>
<font color="Maroon">L0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E53"><i><font color="Green">Lemma118</font></i></a>, <a class="ref" href="euclidlp.html#T15" target="_self">Th15</a></i>;<br/>


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