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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>thus </b><a NAME="E1:110"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="euclidlp.html#R7" target="_self">are_coplane</a>  implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:110_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="euclidlp.html#R7" target="_self">are_coplane</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:110_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#D12" target="_self">Def12</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>take </b>
 <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E4:110_1"/>
(  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:110_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="euclidlp.html#T95" target="_self">Th95</a></i>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:110_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E4:110_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:110_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
 ;<br/><b>thus </b><a NAME="E7:110_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="euclidlp.html#R7" target="_self">are_coplane</a> 
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:110_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="euclidlp.html#E6:110_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="euclidlp.html#D12" target="_self">Def12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:110"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="euclidlp.html#R7" target="_self">are_coplane</a>  )
 ;<br/>


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