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<b>let </b><font color="Maroon" title="c1">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> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">Z</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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>;<br/>



<b>assume </b><a NAME="E1:151"/><i><font color="Green" title="E109">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K26" title="PSCOMP_1:func.26">S-most</a> <font color="Maroon" title="c2">Z</font>
 ;<br/>

<a NAME="E2:151"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">SW-corner</a> <font color="Maroon" title="c2">Z</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">SE-corner</a> <font color="Maroon" title="c2">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Z</font>
 
;<br/>
<a NAME="E3:151"/><b>then </b><i><font color="Green" title="E110">A2</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">SW-corner</a> <font color="Maroon" title="c2">Z</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">SE-corner</a> <font color="Maroon" title="c2">Z</font></span>)</span> &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:151"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">SW-corner</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c2">Z</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K34" title="PSCOMP_1:func.34">S-min</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c2">Z</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K33" title="PSCOMP_1:func.33">S-max</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c2">Z</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">SE-corner</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c2">Z</font> )
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E119"><i><font color="Green" title="E76">Lm11</font></i></a>, <a class="txt" href="pscomp_1.html#E125"><i><font color="Green" title="E82">Lm17</font></i></a>, <a class="txt" href="pscomp_1.html#E186"><i><font color="Green" title="E104">Lm24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:151"/>
<font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K34" title="PSCOMP_1:func.34">S-min</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:151"><i><font color="Green" title="E110">A2</font></i></a>, <a class="ref" href="goboard7.html#T6" title="GOBOARD7:th.6">GOBOARD7:6</a></i>;<br/>

<b>assume </b><a NAME="E6:151"/>
<font color="Maroon" title="c2">Z</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">Z</font> = <font color="Maroon" title="c2">Z</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <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> ;<br/>
<a NAME="E8:151"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K34" title="PSCOMP_1:func.34">S-min</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K26" title="PSCOMP_1:func.26">S-most</a> <font color="Maroon" title="c3">Z</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K33" title="PSCOMP_1:func.33">S-max</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K26" title="PSCOMP_1:func.26">S-most</a> <font color="Maroon" title="c3">Z</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E189"><i><font color="Green" title="E106">Lm25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:151"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K34" title="PSCOMP_1:func.34">S-min</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33" title="PSCOMP_1:func.33">S-max</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:151"><i><font color="Green" title="E109">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E108"><i><font color="Green" title="E73">Lm9</font></i></a></i>;<br/>


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