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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</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> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</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> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="sppol_1.html#R1" title="SPPOL_1:pred.1">is_extremal_in</a> <font color="Maroon" title="c3">P</font>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font>
 ;<br/>

<b>assume </b><a NAME="E3:16"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 ;<br/>

<b>hence </b><a NAME="E4:16"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="sppol_1.html#D1" target="_self" title="SPPOL_1:def.1">SPPOL_1:def 1</a><br/>

<b>let </b><font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c6">p2</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> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E5:16"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>
 ;<br/>

<b>assume </b><a NAME="E6:16"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Q</font>
 ;<br/>

<a NAME="E7:16"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="sppol_1.html#E2:16"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E8:16"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> )
 <b>by </b><i><a class="txt" href="sppol_1.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="sppol_1.html#E5:16"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="sppol_1.html#D1" target="_self" title="SPPOL_1:def.1">Def1</a></i>;<br/>


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