<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="incsp_1.html#NM4">IncSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="incsp_1.html#NM3">PLANE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>3</sub></font>, <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="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:64"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:64"/><i><font color="Green">E42</font></i>: 
not <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a> 
 <b>by </b><i><a class="txt" href="incsp_1.html#E84"><i><font color="Green">Lemma39</font></i></a></i>;<br/>
<a NAME="E4:64"/>
not <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E3:64"><i><font color="Green">E42</font></i></a>, <a class="ref" href="incsp_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E5:64"/><b>then </b>
not <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="enumset1.html#T111">ENUMSET1:111</a></i>;<br/>
<a NAME="E6:64"/><b>then </b>
not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="enumset1.html#T46">ENUMSET1:46</a></i>;<br/>
<a NAME="E7:64"/><b>then </b>
not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E2:64"><i><font color="Green">E41</font></i></a>, <a class="ref" href="incsp_1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E8:64"/><b>then </b>
(  not <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>6</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:64"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E11:64"/><i><font color="Green">E44</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>
<a NAME="E12:64"/>
( not <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a>  &amp; not <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a>  &amp; not <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a>  )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E3:64"><i><font color="Green">E42</font></i></a>, <a class="ref" href="enumset1.html#T110">ENUMSET1:110</a>, <a class="ref" href="enumset1.html#T112">ENUMSET1:112</a>, <a class="ref" href="enumset1.html#T118">ENUMSET1:118</a></i>;<br/>
<a NAME="E13:64"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T37" target="_self">Th37</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:64"/><i><font color="Green">E45</font></i>: 
( ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E10:64"><i><font color="Green">E43</font></i></a></i>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1"/>
<span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a> 
 ;<br/><a NAME="E2:64_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a>  or <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a>  )
 <b>by </b><i><a class="txt" href="incsp_1.html#E3:64"><i><font color="Green">E42</font></i></a>, <a class="txt" href="incsp_1.html#E10:64"><i><font color="Green">E43</font></i></a>, <a class="txt" href="incsp_1.html#E15:64"><i><font color="Green">E45</font></i></a>, <a class="ref" href="enumset1.html#T104">ENUMSET1:104</a></i>;<br/><b>hence </b><a NAME="E3:64_1"/>
contradiction
 <b>by </b><i><a class="txt" href="incsp_1.html#E3:64"><i><font color="Green">E42</font></i></a>, <a class="ref" href="enumset1.html#T105">ENUMSET1:105</a>, <a class="ref" href="enumset1.html#T107">ENUMSET1:107</a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E17:64"/>
not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#NR6" target="_self">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="incsp_1.html#E16:64"><i><font color="Green">E46</font></i></a>, <a class="ref" href="incsp_1.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E18:64"/><b>then </b>
not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#NR6" target="_self">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="enumset1.html#T100">ENUMSET1:100</a></i>;<br/>
<a NAME="E19:64"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E20:64"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R2" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E22:64"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E23:64"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R2" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E24:64"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E2:64"><i><font color="Green">E41</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E25:64"/>
not <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a> 
 
<b>by </b><i><a class="txt" href="incsp_1.html#E16:64"><i><font color="Green">E46</font></i></a>, <a class="ref" href="enumset1.html#T112">ENUMSET1:112</a></i>;<br/>
<a NAME="E26:64"/><b>then </b>
not <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#NR6" target="_self">is_collinear</a> 
 
<b>by </b><i><a class="ref" href="incsp_1.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E27:64"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E28:64"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R2" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E30:64"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E31:64"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E32:64"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R2" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E2:64"><i><font color="Green">E41</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E33:64"/>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E2:64"><i><font color="Green">E41</font></i></a>, <a class="txt" href="incsp_1.html#E24:64"><i><font color="Green">E51</font></i></a>, <a class="txt" href="incsp_1.html#E31:64"><i><font color="Green">E55</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self">Th14</a></i>;<br/>

<b>given </b><font color="Maroon">c<sub>14</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E34:64"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>14</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D6" target="_self">INCSP_1:def 6</a><br/>

<a NAME="E35:64"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E34:64"><i><font color="Green">E57</font></i></a>, <a class="ref" href="incsp_1.html#T12" target="_self">Th12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E37:64"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E38:64"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E50"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E39:64"/>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E34:64"><i><font color="Green">E57</font></i></a>, <a class="ref" href="enumset1.html#T98">ENUMSET1:98</a></i>;<br/>
<a NAME="E40:64"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E34:64"><i><font color="Green">E57</font></i></a>, <a class="ref" href="enumset1.html#T43">ENUMSET1:43</a></i>;<br/>
<a NAME="E41:64"/><b>then </b><i><font color="Green">E61</font></i>: 
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E20:64"><i><font color="Green">E48</font></i></a>, <a class="txt" href="incsp_1.html#E23:64"><i><font color="Green">E50</font></i></a>, <a class="txt" href="incsp_1.html#E28:64"><i><font color="Green">E53</font></i></a>, <a class="txt" href="incsp_1.html#E32:64"><i><font color="Green">E56</font></i></a>, <a class="ref" href="incsp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="incsp_1.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E42:64"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R3" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R3" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E22:64"><i><font color="Green">E49</font></i></a>, <a class="txt" href="incsp_1.html#E30:64"><i><font color="Green">E54</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E43:64"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R2" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R2" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R2" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E35:64"><i><font color="Green">E58</font></i></a>, <a class="txt" href="incsp_1.html#E38:64"><i><font color="Green">E60</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_2"/>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#NR6" target="_self">is_collinear</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:64_2"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E4:64_2"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E3:64_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="enumset1.html#T102">ENUMSET1:102</a></i>;<br/><a NAME="E5:64_2"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E2:64"><i><font color="Green">E41</font></i></a>, <a class="txt" href="incsp_1.html#E24:64"><i><font color="Green">E51</font></i></a>, <a class="ref" href="incsp_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E6:64_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E22:64"><i><font color="Green">E49</font></i></a>, <a class="txt" href="incsp_1.html#E41:64"><i><font color="Green">E61</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E7:64_2"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R2" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E38:64"><i><font color="Green">E60</font></i></a>, <a class="txt" href="incsp_1.html#E4:64_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a>, <a class="ref" href="enumset1.html#T43">ENUMSET1:43</a></i>;<br/><a NAME="E8:64_2"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K7">}</a></span> <a href="incsp_1.html#R4" target="_self">on</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E24:64"><i><font color="Green">E51</font></i></a>, <a class="ref" href="incsp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="incsp_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E9:64_2"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R3" target="_self">on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R1" target="_self">on</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E37:64"><i><font color="Green">E59</font></i></a>, <a class="txt" href="incsp_1.html#E3:64_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a>, <a class="ref" href="incsp_1.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E10:64_2"/>
contradiction
 <b>by </b><i><a class="txt" href="incsp_1.html#E11:64"><i><font color="Green">E44</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E45:64"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R2" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="incsp_1.html#R2" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E19:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="incsp_1.html#E27:64"><i><font color="Green">E52</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E46:64"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E23:64"><i><font color="Green">E50</font></i></a>, <a class="txt" href="incsp_1.html#E43:64"><i><font color="Green">E62</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E47:64"/><b>then </b>
 <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E44:64"><i><font color="Green">E63</font></i></a>, <a class="ref" href="incsp_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E48:64"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R2" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E27:64"><i><font color="Green">E52</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E49:64"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K8">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E19:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="incsp_1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E50:64"/><b>then </b>
<span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#R5" target="_self">on</a>  <a href="incsp_1.html#K2" target="_self">Plane</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="enumset1.html#T46">ENUMSET1:46</a></i>;<br/>
<a NAME="E51:64"/><b>then </b>
<span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K9">}</a></span> <a href="incsp_1.html#NR7" target="_self">is_coplanar</a> 
 
<b>by </b><i><a class="ref" href="incsp_1.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E52:64"/>
contradiction
 <b>by </b><i><a class="txt" href="incsp_1.html#E16:64"><i><font color="Green">E46</font></i></a>, <a class="ref" href="enumset1.html#T110">ENUMSET1:110</a></i>;<br/>


</div>
