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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font>, <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/>(  not <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">Q</font> or  for <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>  holds <br/>(  not <font color="Olive" title="b3">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P</font> or  not <font color="Olive" title="b3">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b2">Q</font> ) or  ex <font color="Olive" title="b3">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 <br/> for <font color="Olive" title="b4">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( ( <font color="Olive" title="b4">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b4">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b2">Q</font> ) iff <font color="Olive" title="b4">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">L</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Q</font> or  for <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>  holds <br/>(  not <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> or  not <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> ) or  ex <font color="Olive" title="b1">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 <br/> for <font color="Olive" title="b2">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( ( <font color="Olive" title="b2">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b2">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> ) iff <font color="Olive" title="b2">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:77"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  for <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>  holds <br/>(  not <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> or  not <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> ) or  ex <font color="Olive" title="b1">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 <br/> for <font color="Olive" title="b2">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( ( <font color="Olive" title="b2">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b2">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> ) iff <font color="Olive" title="b2">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L</font> ) )</span><br/>

<span class="kw">given </span><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><span class="kw"> such that </span><a NAME="E3:77"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">and </span><br/><a NAME="E4:77"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/> for <font color="Olive" title="b2">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( ( <font color="Olive" title="b2">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b2">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> ) iff <font color="Olive" title="b2">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">L</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">C</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E6:77"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">C</font>
 <span class="kw">and </span><br/><a NAME="E7:77"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
<font color="Maroon" title="c5">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">and </span><br/><a NAME="E8:77"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c5">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="incsp_1.html#D15" target="_self" title="INCSP_1:def.15">Def15</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c6">L</font> =  <a href="incsp_1.html#K1" title="INCSP_1:func.1">Line</a> (<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">C</font>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( ( <font color="Olive" title="b1">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> ) iff <font color="Olive" title="b1">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> )</span><br/>

<a NAME="E9:77"/><span class="lab"><font color="Green" title="E61">A7</font></span>: 
<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">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="c6">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:77"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="ref" href="incsp_1.html#D19" target="_self" title="INCSP_1:def.19">Def19</a></span>;<br/>
<a NAME="E10:77"/>
<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">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="c3">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="txt" href="#E8:77"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="ref" href="incsp_1.html#T3" target="_self" title="INCSP_1:th.3">Th13</a></span>;<br/>
<a NAME="E11:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A8</font></span>: 
<font color="Maroon" title="c6">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c3">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:77"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E9:77"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">B</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> ) iff <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> )</span><br/>

<a NAME="E12:77"/>
<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">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="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E7:77"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="incsp_1.html#T3" target="_self" title="INCSP_1:th.3">Th13</a></span>;<br/>
<a NAME="E13:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A9</font></span>: 
<font color="Maroon" title="c6">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:77"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E9:77"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="ref" href="incsp_1.html#D14" target="_self" title="INCSP_1:def.14">Def14</a></span>;<br/>
<span class="kw">thus </span><a NAME="E14:77"/>
( <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> implies <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> implies ( <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:77_1"/><span class="lab"><font color="Green" title="E64">A10</font></span>: 
<font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">and </span><br/><a NAME="E2:77_1"/><span class="lab"><font color="Green" title="E65">A11</font></span>: 
<font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font>
 <span class="kw">and </span><br/><a NAME="E3:77_1"/><span class="lab"><font color="Green" title="E66">A12</font></span>: 
 not <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">P1</font> being   <a href="incsp_1.html#NM3" title="INCSP_1:NM.3">PLANE</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E5:77_1"/><span class="lab"><font color="Green" title="E67">A13</font></span>: 
 for <font color="Olive" title="b1">P2</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="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P2</font> &amp; <font color="Maroon" title="c6">L</font> <a href="incsp_1.html#R3" title="INCSP_1:pred.3">on</a> <font color="Olive" title="b1">P2</font> ) iff <font color="Maroon" title="c8">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="ref" href="incsp_1.html#T26" target="_self" title="INCSP_1:th.26">Th48</a></span>;<br/>
<a NAME="E6:77_1"/>
<font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:77"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="txt" href="#E1:77_1"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="txt" href="#E5:77_1"><span class="lab"><font color="Green" title="E67">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:77_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E11:77"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="txt" href="#E2:77_1"><span class="lab"><font color="Green" title="E65">A11</font></span></a>, <a class="txt" href="#E5:77_1"><span class="lab"><font color="Green" title="E67">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E15:77"/>
( <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">L</font> implies ( <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c7">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c3">Q</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:77"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="txt" href="#E11:77"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="incsp_1.html#D17" target="_self" title="INCSP_1:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
