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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">L1</font>, <font color="Maroon" title="c3">L2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font>;<br/>





<b>assume </b><a NAME="E1:112"/><i><font color="Green" title="E89">A1</font></i>: 
( <font color="Maroon" title="c2">L1</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  &amp; <font color="Maroon" title="c3">L2</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  &amp; <font color="Maroon" title="c2">L1</font>,<font color="Maroon" title="c3">L2</font> <a href="euclidlp.html#R7" title="EUCLIDLP:pred.7">are_coplane</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font>, <font color="Maroon" title="c6">x3</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:112"/><i><font color="Green" title="E90">A2</font></i>: 
( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#D12" target="_self" title="EUCLIDLP:def.12">Def12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">P</font> =  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font>;<br/>
<b>assume </b><a NAME="E4:112"/><i><font color="Green" title="E91">A3</font></i>: 
<font color="Maroon" title="c2">L1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">L2</font>
 ;<br/>

<a NAME="E5:112"/><i><font color="Green" title="E92">A4</font></i>: 
 <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclidlp.html#K5" title="EUCLIDLP:func.5">plane_of_REAL</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c8">y0</font>, <font color="Maroon" title="c9">y1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E7:112"/><i><font color="Green" title="E93">A5</font></i>: 
( <font color="Maroon" title="c8">y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y1</font> &amp; <font color="Maroon" title="c2">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c8">y0</font>,<font color="Maroon" title="c9">y1</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:112"><i><font color="Green" title="E89">A1</font></i></a>, <a class="ref" href="euclid_4.html#D2" title="EUCLID_4:def.2">EUCLID_4:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">y2</font>, <font color="Maroon" title="c11">y3</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E9:112"/><i><font color="Green" title="E94">A6</font></i>: 
( <font color="Maroon" title="c10">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y3</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c10">y2</font>,<font color="Maroon" title="c11">y3</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:112"><i><font color="Green" title="E89">A1</font></i></a>, <a class="ref" href="euclid_4.html#D2" title="EUCLID_4:def.2">EUCLID_4:def 2</a></i>;<br/>
<a NAME="E10:112"/><i><font color="Green" title="E95">A7</font></i>: 
( <font color="Maroon" title="c8">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E7:112"><i><font color="Green" title="E93">A5</font></i></a>, <a class="txt" href="euclidlp.html#E9:112"><i><font color="Green" title="E94">A6</font></i></a>, <a class="ref" href="euclid_4.html#T10" title="EUCLID_4:th.10">EUCLID_4:10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E12:112"/><i><font color="Green" title="E96">A8</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c8">y0</font>,<font color="Maroon" title="c9">y1</font> &amp; <font color="Maroon" title="c8">y0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y1</font>,<font color="Maroon" title="c10">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c12">y</font> <a href="euclid_4.html#R1" title="EUCLID_4:pred.1">are_orthogonal</a>  )
 <b>by </b><i><a class="ref" href="euclidlp.html#T48" target="_self" title="EUCLIDLP:th.48">Th48</a></i>;<br/>
<a NAME="E13:112"/><i><font color="Green" title="E97">A9</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y2</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E4:112"><i><font color="Green" title="E91">A3</font></i></a>, <a class="txt" href="euclidlp.html#E7:112"><i><font color="Green" title="E93">A5</font></i></a>, <a class="txt" href="euclidlp.html#E10:112"><i><font color="Green" title="E95">A7</font></i></a>, <a class="txt" href="euclidlp.html#E12:112"><i><font color="Green" title="E96">A8</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E14:112"/><b>then </b>
( <font color="Maroon" title="c8">y0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E7:112"><i><font color="Green" title="E93">A5</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></i>;<br/>
<a NAME="E15:112"/><b>then </b>
<font color="Maroon" title="c8">y0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y1</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c10">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c12">y</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E12:112"><i><font color="Green" title="E96">A8</font></i></a>, <a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></i>;<br/>
<a NAME="E16:112"/><b>then </b><i><font color="Green" title="E98">A10</font></i>: 
<font color="Maroon" title="c8">y0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y1</font>,<font color="Maroon" title="c10">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c12">y</font> <a href="euclidlp.html#R3" title="EUCLIDLP:pred.3">are_lindependent2</a> 
 
<b>by </b><i><a class="ref" href="euclidlp.html#T50" target="_self" title="EUCLIDLP:th.50">Th50</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">y'</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E18:112"/><i><font color="Green" title="E99">A11</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">y'</font> &amp; <font color="Maroon" title="c13">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:112"><i><font color="Green" title="E89">A1</font></i></a>, <a class="ref" href="euclid_4.html#T15" title="EUCLID_4:th.15">EUCLID_4:15</a></i>;<br/>
<a NAME="E19:112"/>
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c10">y2</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c10">y2</font> )
 
<b>by </b><i><a class="ref" href="euclid_4.html#T10" title="EUCLID_4:th.10">EUCLID_4:10</a></i>;<br/>
<a NAME="E20:112"/><b>then </b><i><font color="Green" title="E100">A12</font></i>: 
<font color="Maroon" title="c13">y'</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c10">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c12">y</font> <a href="euclidlp.html#R3" title="EUCLIDLP:pred.3">are_lindependent2</a> 
 
<b>by </b><i><a class="txt" href="euclidlp.html#E7:112"><i><font color="Green" title="E93">A5</font></i></a>, <a class="txt" href="euclidlp.html#E12:112"><i><font color="Green" title="E96">A8</font></i></a>, <a class="txt" href="euclidlp.html#E13:112"><i><font color="Green" title="E97">A9</font></i></a>, <a class="txt" href="euclidlp.html#E16:112"><i><font color="Green" title="E98">A10</font></i></a>, <a class="txt" href="euclidlp.html#E18:112"><i><font color="Green" title="E99">A11</font></i></a>, <a class="ref" href="euclidlp.html#T45" target="_self" title="EUCLIDLP:th.45">Th45</a></i>;<br/>
<a NAME="E21:112"/><b>then </b><i><font color="Green" title="E101">A13</font></i>: 
 <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">y'</font>,<font color="Maroon" title="c10">y2</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E3:112"><i><font color="Green" title="E90">A2</font></i></a>, <a class="txt" href="euclidlp.html#E5:112"><i><font color="Green" title="E92">A4</font></i></a>, <a class="txt" href="euclidlp.html#E7:112"><i><font color="Green" title="E93">A5</font></i></a>, <a class="txt" href="euclidlp.html#E10:112"><i><font color="Green" title="E95">A7</font></i></a>, <a class="txt" href="euclidlp.html#E12:112"><i><font color="Green" title="E96">A8</font></i></a>, <a class="txt" href="euclidlp.html#E18:112"><i><font color="Green" title="E99">A11</font></i></a>, <a class="ref" href="euclidlp.html#T97" target="_self" title="EUCLIDLP:th.97">Th97</a></i>;<br/>
<b>take </b>
 <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font>
;<br/>

<b>thus </b><a NAME="E22:112"/>
(  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K5" title="EUCLIDLP:func.5">plane_of_REAL</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font> &amp;  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font> is <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:112"><i><font color="Green" title="E90">A2</font></i></a>, <a class="txt" href="euclidlp.html#E5:112"><i><font color="Green" title="E92">A4</font></i></a>, <a class="txt" href="euclidlp.html#E20:112"><i><font color="Green" title="E100">A12</font></i></a>, <a class="txt" href="euclidlp.html#E21:112"><i><font color="Green" title="E101">A13</font></i></a>, <a class="ref" href="euclidlp.html#D10" target="_self" title="EUCLIDLP:def.10">Def10</a></i>;<br/>


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