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<b>let </b><font color="Maroon" title="c1">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <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">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b3">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b4">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b3">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">q2</font> holds <br/><font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font></span><br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q1</font>, <font color="Maroon" title="c5">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q2</font> implies <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:83"/><i><font color="Green" title="E67">A1</font></i>: 
<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font>
 <b>and </b><br/><a NAME="E2:83"/><i><font color="Green" title="E68">A2</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>and </b><br/><a NAME="E3:83"/><i><font color="Green" title="E69">A3</font></i>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>and </b><br/><a NAME="E4:83"/><i><font color="Green" title="E70">A4</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font></span><br/>

<a NAME="E5:83"/><i><font color="Green" title="E71">A5</font></i>: 
<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E1:83"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="sppol_2.html#T54" target="_self" title="SPPOL_2:th.54">Th54</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q1</font> or <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font> or <font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q1</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1"><b>suppose </b></a><a NAME="E1:83_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font></span><br/><div class="add"><b>hence </b><a NAME="E2:83_1_1"/>
<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:83"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="sppol_2.html#E3:83"><i><font color="Green" title="E69">A3</font></i></a>, <a class="txt" href="sppol_2.html#E4:83"><i><font color="Green" title="E70">A4</font></i></a>, <a class="ref" href="sppol_2.html#T55" target="_self" title="SPPOL_2:th.55">Th55</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_2"><b>suppose </b></a><a NAME="E1:83_1_2"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font></span><br/><div class="add"><b>hence </b><a NAME="E2:83_1_2"/>
<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:83"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="sppol_2.html#E4:83"><i><font color="Green" title="E70">A4</font></i></a>, <a class="txt" href="sppol_2.html#E5:83"><i><font color="Green" title="E71">A5</font></i></a>, <a class="ref" href="sppol_2.html#T56" target="_self" title="SPPOL_2:th.56">Th56</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_3"><b>suppose </b></a><a NAME="E1:83_1_3"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:83_1_3"/><b>then </b>
<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:83"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="sppol_2.html#E2:83"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="sppol_2.html#T55" target="_self" title="SPPOL_2:th.55">Th55</a></i>;<br/><a NAME="E3:83_1_3"/><b>then </b>
<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:83"><i><font color="Green" title="E69">A3</font></i></a>, <a class="txt" href="sppol_2.html#E4:83"><i><font color="Green" title="E70">A4</font></i></a>, <a class="ref" href="sppol_2.html#T56" target="_self" title="SPPOL_2:th.56">Th56</a></i>;<br/><b>hence </b><a NAME="E4:83_1_3"/>
<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T54" target="_self" title="SPPOL_2:th.54">Th54</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_4"><b>suppose </b></a><a NAME="E1:83_1_4"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:83_1_4"/><b>then </b>
<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c3">p2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:83"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="sppol_2.html#E2:83"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="sppol_2.html#T56" target="_self" title="SPPOL_2:th.56">Th56</a></i>;<br/><b>hence </b><a NAME="E3:83_1_4"/>
<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="sppol_2.html#R1" title="SPPOL_2:pred.1">split</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E3:83"><i><font color="Green" title="E69">A3</font></i></a>, <a class="txt" href="sppol_2.html#E4:83"><i><font color="Green" title="E70">A4</font></i></a>, <a class="ref" href="sppol_2.html#T55" target="_self" title="SPPOL_2:th.55">Th55</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></div><b>end;</b></div>

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