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<b>let </b><font color="Maroon" title="c1">S</font> be    <a href="incsp_1.html#L2" title="INCSP_1:struct.2">IncStruct</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">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font>, <font color="Olive" title="b4">D</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b5">P</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/> ( <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font>,<font color="Olive" title="b4">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b5">P</font> iff ( <font color="Olive" title="b1">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b5">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="Olive" title="b5">P</font> &amp; <font color="Olive" title="b3">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b5">P</font> &amp; <font color="Olive" title="b4">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b5">P</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font>, <font color="Maroon" title="c5">D</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <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> holds <br/> ( <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Olive" title="b1">P</font> iff ( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">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="Maroon" title="c4">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c5">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Olive" title="b1">P</font> ) )</span><br/><b>let </b><font color="Maroon" title="c6">P</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"> ( <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c6">P</font> iff ( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c3">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c4">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> ) )</span><br/>











<b>thus </b><a NAME="E1:18"/>
( <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c6">P</font> implies ( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c3">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c4">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">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">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c3">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c4">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> implies <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c6">P</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E13">A1</font></i>: 
<span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c6">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">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c3">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c4">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> )</span><br/>

<a NAME="E2:18_1"/>
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> &amp; <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> &amp; <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> )
 
<b>by </b><i><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:18_1"/>
( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c3">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c4">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> )
 <b>by </b><i><a class="txt" href="incsp_1.html#E1:18_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="incsp_1.html#D5" target="_self" title="INCSP_1:def.5">Def5</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>assume </b><a NAME="E2:18"/><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c3">B</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c4">C</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">D</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> <a href="incsp_1.html#R5" title="INCSP_1:pred.5">on</a> <font color="Maroon" title="c6">P</font></span><br/>

<b>let </b><font color="Maroon" title="c7">E</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="incsp_1.html#D5" target="_self" title="INCSP_1:def.5">INCSP_1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> implies <font color="Maroon" title="c7">E</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font> )</span><br/>

<b>assume </b><a NAME="E3:18"/>
<font color="Maroon" title="c7">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">E</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font></span><br/>

<b>hence </b><a NAME="E4:18"/>
<font color="Maroon" title="c7">E</font> <a href="incsp_1.html#R2" title="INCSP_1:pred.2">on</a> <font color="Maroon" title="c6">P</font>
 <b>by </b><i><a class="txt" href="incsp_1.html#E2:18"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</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>
