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<div class="add">

<b>let </b><font color="Maroon" title="c1">S</font> be   <a href="incsp_1.html#NM4" title="INCSP_1:NM.4">IncSpace</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">K</font>, <font color="Olive" title="b2">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">L</font> &amp;  ex <font color="Olive" title="b3">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b3">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b3">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">L</font> ) holds <br/> ex <font color="Olive" title="b3">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font> st <br/>for <font color="Olive" title="b4">Q</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( ( <font color="Olive" title="b1">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b4">Q</font> &amp; <font color="Olive" title="b2">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b4">Q</font> ) iff <font color="Olive" title="b3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Q</font> )</span><br/><b>let </b><font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c3">L</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">S</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">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L</font> &amp;  ex <font color="Olive" title="b1">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">L</font> ) implies  ex <font color="Olive" title="b1">P</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font> st <br/>for <font color="Olive" title="b2">Q</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( ( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">Q</font> ) iff <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font> ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L</font>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E42">A2</font></i>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">L</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="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font> st <br/>for <font color="Olive" title="b2">Q</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( ( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b2">Q</font> ) iff <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c4">A</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E4:48"/><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font>
 <b>and </b><br/><a NAME="E5:48"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E2:48"><i><font color="Green" title="E42">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E7:48"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font>
 <b>and </b><br/><a NAME="E8:48"/><i><font color="Green" title="E46">A6</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E60"><i><font color="Green" title="E36">Lm1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">C</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E10:48"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">C</font>
 <b>and </b><br/><a NAME="E11:48"/><i><font color="Green" title="E48">A8</font></i>: 
<font color="Maroon" title="c6">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E60"><i><font color="Green" title="E36">Lm1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">P</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E13:48"/><i><font color="Green" title="E49">A9</font></i>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c7">P</font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D12" target="_self" title="INCSP_1:def.12">Def12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">P</font>
; <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">Q</font> being  <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( ( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">Q</font> ) iff <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> )</span><br/>

<b>let </b><font color="Maroon" title="c8">Q</font> be   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</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">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> ) iff <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font> )</span><br/>

<b>thus </b><a NAME="E14:48"/>
( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> implies <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font> implies ( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green" title="E50">A10</font></i>: 
( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font></span><br/>

<a NAME="E2:48_1"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E5:48"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="incsp_1.html#E11:48"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/>
<a NAME="E3:48_1"/><b>then </b>
not <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="incsp_1.html#E1:48"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="incsp_1.html#E10:48"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="incsp_1.html#D10" target="_self" title="INCSP_1:def.10">Def10</a></i>;<br/>
<a NAME="E4:48_1"/><b>then </b>
( not <font color="Maroon" title="c6">C</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c2">K</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E4:48"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="incsp_1.html#E8:48"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a></i>;<br/>
<a NAME="E5:48_1"/><b>then </b><i><font color="Green" title="E51">A11</font></i>: 
not <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#NR6" title="INCSP_1:NR.6">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="incsp_1.html#E7:48"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="incsp_1.html#T39" target="_self" title="INCSP_1:th.39">Th39</a></i>;<br/>
<a NAME="E6:48_1"/>
( <font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c6">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c8">Q</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E4:48"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="incsp_1.html#E8:48"><i><font color="Green" title="E46">A6</font></i></a>, <a class="txt" href="incsp_1.html#E11:48"><i><font color="Green" title="E48">A8</font></i></a>, <a class="txt" href="incsp_1.html#E1:48_1"><i><font color="Green" title="E50">A10</font></i></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></i>;<br/>
<a NAME="E7:48_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c8">Q</font>
 
<b>by </b><i><a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E8:48_1"/>
<font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E13:48"><i><font color="Green" title="E49">A9</font></i></a>, <a class="txt" href="incsp_1.html#E5:48_1"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="incsp_1.html#D13" target="_self" title="INCSP_1:def.13">Def13</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>

<a NAME="E15:48"/>
( <font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c5">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c6">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c7">P</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E13:48"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="incsp_1.html#T14" target="_self" title="INCSP_1:th.14">Th14</a></i>;<br/>
<a NAME="E16:48"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c2">K</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c3">L</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c7">P</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c7">P</font> )
 
<b>by </b><i><a class="txt" href="incsp_1.html#E4:48"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="incsp_1.html#E5:48"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="incsp_1.html#E8:48"><i><font color="Green" title="E46">A6</font></i></a>, <a class="txt" href="incsp_1.html#E11:48"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="incsp_1.html#T11" target="_self" title="INCSP_1:th.11">Th11</a>, <a class="ref" href="incsp_1.html#T13" target="_self" title="INCSP_1:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E17:48"/>
( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font> implies ( <font color="Maroon" title="c2">K</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c3">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c8">Q</font> ) )
 <b>by </b><i><a class="txt" href="incsp_1.html#E7:48"><i><font color="Green" title="E45">A5</font></i></a>, <a class="txt" href="incsp_1.html#E10:48"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></i>; <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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