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<b>let </b><font color="Maroon" title="c1">X</font> be  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="E1:131"/><i><font color="Green" title="E77">A1</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</a> <font color="Maroon" title="c1">X</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#K8" title="PSCOMP_1:func.8">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">proj1</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<b>set </b><font color="Maroon" title="c2">LX</font> =  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">N-most</a> <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E2:131"/><i><font color="Green" title="E78">A2</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <font color="Maroon" title="c1">X</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#K8" title="PSCOMP_1:func.8">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">proj1</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">N-most</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E3:131"/><i><font color="Green" title="E79">A3</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28" title="PSCOMP_1:func.28">N-max</a> <font color="Maroon" title="c1">X</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#K9" title="PSCOMP_1:func.9">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">proj1</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">N-most</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E4:131"/><i><font color="Green" title="E80">A4</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">NE-corner</a> <font color="Maroon" title="c1">X</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#K9" title="PSCOMP_1:func.9">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">proj1</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E5:131"/><i><font color="Green" title="E81">A5</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</a> <font color="Maroon" title="c1">X</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <font color="Maroon" title="c1">X</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#E1:131"><i><font color="Green" title="E77">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E2:131"><i><font color="Green" title="E78">A2</font></i></a>, <a class="ref" href="pscomp_1.html#T62" target="_self" title="PSCOMP_1:th.62">Th62</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:131"/><i><font color="Green" title="E82">A6</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <font color="Maroon" title="c1">X</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K28" title="PSCOMP_1:func.28">N-max</a> <font color="Maroon" title="c1">X</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#E2:131"><i><font color="Green" title="E78">A2</font></i></a>, <a class="txt" href="pscomp_1.html#E3:131"><i><font color="Green" title="E79">A3</font></i></a>, <a class="ref" href="pscomp_1.html#T53" target="_self" title="PSCOMP_1:th.53">Th53</a></i>;<br/>
<a NAME="E7:131"/><i><font color="Green" title="E83">A7</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28" title="PSCOMP_1:func.28">N-max</a> <font color="Maroon" title="c1">X</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">NE-corner</a> <font color="Maroon" title="c1">X</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#E3:131"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="pscomp_1.html#E4:131"><i><font color="Green" title="E80">A4</font></i></a>, <a class="ref" href="pscomp_1.html#T63" target="_self" title="PSCOMP_1:th.63">Th63</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>thus </b><a NAME="E8:131"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</a> <font color="Maroon" title="c1">X</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <font color="Maroon" title="c1">X</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#E1:131"><i><font color="Green" title="E77">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E2:131"><i><font color="Green" title="E78">A2</font></i></a>, <a class="ref" href="pscomp_1.html#T62" target="_self" title="PSCOMP_1:th.62">Th62</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>

<b>thus </b><a NAME="E9:131"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</a> <font color="Maroon" title="c1">X</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K28" title="PSCOMP_1:func.28">N-max</a> <font color="Maroon" title="c1">X</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#E5:131"><i><font color="Green" title="E81">A5</font></i></a>, <a class="txt" href="pscomp_1.html#E6:131"><i><font color="Green" title="E82">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>

<b>hence </b><a NAME="E10:131"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">NW-corner</a> <font color="Maroon" title="c1">X</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">NE-corner</a> <font color="Maroon" title="c1">X</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#E7:131"><i><font color="Green" title="E83">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>

<b>thus </b><a NAME="E11:131"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <font color="Maroon" title="c1">X</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K28" title="PSCOMP_1:func.28">N-max</a> <font color="Maroon" title="c1">X</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#E2:131"><i><font color="Green" title="E78">A2</font></i></a>, <a class="txt" href="pscomp_1.html#E3:131"><i><font color="Green" title="E79">A3</font></i></a>, <a class="ref" href="pscomp_1.html#T53" target="_self" title="PSCOMP_1:th.53">Th53</a></i>;<br/>

<b>thus </b><a NAME="E12:131"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <font color="Maroon" title="c1">X</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">NE-corner</a> <font color="Maroon" title="c1">X</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#E6:131"><i><font color="Green" title="E82">A6</font></i></a>, <a class="txt" href="pscomp_1.html#E7:131"><i><font color="Green" title="E83">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>

<b>thus </b><a NAME="E13:131"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28" title="PSCOMP_1:func.28">N-max</a> <font color="Maroon" title="c1">X</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">NE-corner</a> <font color="Maroon" title="c1">X</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#E3:131"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="pscomp_1.html#E4:131"><i><font color="Green" title="E80">A4</font></i></a>, <a class="ref" href="pscomp_1.html#T63" target="_self" title="PSCOMP_1:th.63">Th63</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>


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