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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">L</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="E1:125"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">L</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> )
 ;<br/>

<a NAME="E2:125"/><b>then </b><i><font color="Green">E66</font></i>: 
( <font color="Maroon">L</font>,<font color="Maroon">L1</font> <a href="euclidlp.html#R7" target="_self">are_coplane</a>  &amp; <font color="Maroon">L</font>,<font color="Maroon">L2</font> <a href="euclidlp.html#R7" target="_self">are_coplane</a>  &amp; <font color="Maroon">L1</font>,<font color="Maroon">L2</font> <a href="euclidlp.html#R7" target="_self">are_coplane</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:125"><i><font color="Green">E90</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:125"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">L</font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">L</font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">L2</font> )
 ;<br/>

<a NAME="E4:125"/><b>then </b><i><font color="Green">E89</font></i>: 
( <font color="Maroon">L</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">L</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">L2</font> )
 
<b>by </b><i>, <a class="ref" href="euclidlp.html#T41" target="_self">Th41</a></i>;<br/>
<b>then 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="E6:125"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</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#T3" target="_self">Th3</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="E8:125"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E9:125"><i><font color="Green">E106</font></i></a>, <a class="ref" href="euclidlp.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E9:125"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">L</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#E8:125"><i><font color="Green">E105</font></i></a>, <a class="txt" href="euclidlp.html#E15"><i><font color="Green">Lemma69</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:125_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> or <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1_1_1"><b>case </b></a><a NAME="E1:125_1_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/><div class="add"><a NAME="E2:125_1_1_1"/>
<font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font>
 <b>by </b><i>, <a class="txt" href="euclidlp.html#E8:125"><i><font color="Green">E105</font></i></a>, <a class="txt" href="euclidlp.html#E1:125_1_1_1"><i><font color="Green">E107</font></i></a>, , , <a class="ref" href="euclidlp.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E3:125_1_1_1"/>
<font color="Maroon">L1</font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">L2</font>
 <b>by </b><i>, <a class="ref" href="euclidlp.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1_1_2"><b>case </b></a><a NAME="E1:125_1_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font>
 ;<br/><div class="add"><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="E3:125_1_1_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> )
 <b>by </b><i>, <a class="txt" href="euclidlp.html#E3:125_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>;<br/><b>consider </b><font color="Maroon">x</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="E5:125_1_1_2"/><i><font color="Green">E110</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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:125_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E3:125_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E6:125_1_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E8:125"><i><font color="Green">E105</font></i></a>, , <a class="txt" href="euclidlp.html#E1:125_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E5:125_1_1_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="euclidlp.html#E6:125_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="euclidlp.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E7:125_1_1_2"/><b>then </b>
<font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a> 
 <b>by </b><i/>;<br/><a NAME="E8:125_1_1_2"/><b>then </b><i><font color="Green">E112</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">x1</font>,<font color="Maroon">x0</font>,<font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="txt" href="euclidlp.html#E1:125_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E6:125_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="euclidlp.html#T33" target="_self">Th33</a></i>;<br/><b>consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a3</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:125_1_1_2"/><i><font color="Green">E113</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a3</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x</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">x1</font></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">x0</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E8:125_1_1_2"><i><font color="Green">E112</font></i></a>, , <a class="txt" href="euclidlp.html#E11:125_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="euclidlp.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E11:125_1_1_2"/><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="euclidlp.html#T42" target="_self">Th42</a></i>;<br/><i><font color="Green">E151</font></i>: <font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font> = 
<span class="p1">(<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> <font color="Maroon">x2</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</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">x0</font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font>
<b>by </b><i>, <a class="ref" href="euclidlp.html#T42" target="_self">Th42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</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">a2</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</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">x0</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T45" target="_self">Th45</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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">x2</font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">x2</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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">a1</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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="E13:125_1_1_2"/>
<font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E8:125"><i><font color="Green">E105</font></i></a>, , <a class="txt" href="euclidlp.html#E1:125_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E5:125_1_1_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="euclidlp.html#E8:125_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="euclidlp.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E14:125_1_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font>,<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font> <a href="euclid_4.html#R1">are_orthogonal</a> 
 <b>by </b><i/>;<br/><a NAME="E15:125_1_1_2"/>
<font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font> <a href="euclid_4.html#R1">are_orthogonal</a> 
 <b>by </b><i><a class="txt" href="euclidlp.html#E10:125_1_1_2"><i><font color="Green">E113</font></i></a>, </i>;<br/><a NAME="E16:125_1_1_2"/><b>then </b><i><font color="Green">E153</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">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</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/><i><font color="Green">E154</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">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> = 
<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">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i/>
<br/>.= 
<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">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</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"><a href="euclid.html#K6">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i/>
<br/>.= 
<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">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i/>
<br/>.= 
<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">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T6">EUCLID_4:6</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">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <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">x1</font></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <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">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T27">EUCLID_4:27</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="euclidlp.html#T48" target="_self">Th48</a></i>
;<br/><i><font color="Green">E155</font></i>: <font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font> = 
<font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
<br/>.= 
<font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E156</font></i>: 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">a1</font> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="txt" href="euclidlp.html#E53"><i><font color="Green">Lemma118</font></i></a>, <a class="ref" href="euclid_4.html#D6">EUCLID_4:def 6</a>, <a class="txt" href="euclidlp.html#E51"><i><font color="Green">Lemma117</font></i></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">a1</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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">a1</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">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</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">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</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">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T26">EUCLID_4:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</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">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K3">+</a> <span class="p3"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T25">EUCLID_4:25</a>, <a class="ref" href="euclidlp.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a3</font></span>)</span> <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">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i/>
;<br/><a NAME="E20:125_1_1_2"/>
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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#E5:125_1_1_2"><i><font color="Green">E110</font></i></a>, </i>;<br/><a NAME="E21:125_1_1_2"/><b>then </b>
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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/><a NAME="E22:125_1_1_2"/><b>then </b>
<font color="Maroon">a1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="euclidlp.html#T49" target="_self">Th49</a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E23:125_1_1_2"/><b>then </b>
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">a1</font>
 ;<br/><b>then </b><i><font color="Green">E157</font></i>: <font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font> = 
<font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="euclidlp.html#E51"><i><font color="Green">Lemma117</font></i></a></i>
<br/>.= 
<font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span> <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">x2</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/><a NAME="E25:125_1_1_2"/>
( <font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</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">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</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#E6:125_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="euclidlp.html#E8:125_1_1_2"><i><font color="Green">E112</font></i></a>, </i>;<br/><a NAME="E26:125_1_1_2"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T50" target="_self">Th50</a>, </i>;<br/><a NAME="E27:125_1_1_2"/>
( <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x0</font>,<font color="Maroon">x1</font> &amp; <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x2</font>,<font color="Maroon">x</font> )
 <b>by </b><i>, <a class="txt" href="euclidlp.html#E1:125_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="euclidlp.html#E6:125_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="euclidlp.html#E8:125_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="euclidlp.html#E2:125"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E28:125_1_1_2"/>
<font color="Maroon">L1</font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">L2</font>
 <b>by </b><i>, <a class="txt" href="euclidlp.html#E8"><i><font color="Green">Lemma60</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:125"/>
<font color="Maroon">L1</font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">L2</font>
 ;<br/>


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