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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L1</font>, <font color="Olive" title="b2">L2</font> being   <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> holds <br/> ( <font color="Olive" title="b1">L1</font>,<font color="Olive" title="b2">L2</font> <a href="euclidlp.html#R7" title="EUCLIDLP:pred.7">are_coplane</a>  iff  ex <font color="Olive" title="b3">P</font> being   <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> st <br/>( <font color="Olive" title="b1">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">P</font> ) )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  iff  ex <font color="Olive" title="b1">P</font> being   <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> st <br/>( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> ) )</span><br/>





<b>thus </b><a NAME="E1:109"/>
( <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>  implies  ex <font color="Olive" title="b1">P</font> being   <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> st <br/>( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">P</font> being   <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> st <br/>( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> ) implies <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>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:109_1"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being   <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> st <br/>( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> )</span><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="E3:109_1"/><i><font color="Green" title="E85">A1</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>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>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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> )</span><br/>

<b>thus </b><a NAME="E4:109_1"/>
(  <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> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:109_1"><i><font color="Green" title="E85">A1</font></i></a>, <a class="ref" href="euclidlp.html#T95" target="_self" title="EUCLIDLP:th.95">Th95</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_2"/>
 ex <font color="Olive" title="b1">P</font> being   <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> st <br/>( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> </span><br/><b>then consider </b><font color="Maroon" title="c4">P</font> being    <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><b> such that </b><br/><a NAME="E3:109_2"/><i><font color="Green" title="E85">A2</font></i>: 
( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P</font> )
 ;<br/><a NAME="E4:109_2"/>
<font color="Maroon" title="c4">P</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>then consider </b><font color="Maroon" title="c5">P'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E6:109_2"/><i><font color="Green" title="E86">A3</font></i>: 
( <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P'</font> &amp;  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">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> st <font color="Maroon" title="c5">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font> )
 ;<br/><b>thus </b><a NAME="E7:109_2"/>
<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> 
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:109_2"><i><font color="Green" title="E85">A2</font></i></a>, <a class="txt" href="euclidlp.html#E6:109_2"><i><font color="Green" title="E86">A3</font></i></a>, <a class="ref" href="euclidlp.html#D12" target="_self" title="EUCLIDLP:def.12">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:109"/>
(  ex <font color="Olive" title="b1">P</font> being   <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> st <br/>( <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> ) implies <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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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