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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:80"/><i><font color="Green">E60</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E2:80"/><i><font color="Green">E61</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E3:80"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E4:80"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E5:80"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E6:80"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E7:80"/><i><font color="Green">E66</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_1.html#E4:80"><i><font color="Green">E63</font></i></a>, <a class="txt" href="sprect_1.html#E5:80"><i><font color="Green">E64</font></i></a>, <a class="ref" href="sprect_1.html#T62" target="_self">Th62</a></i>;<br/>
<a NAME="E8:80"/><i><font color="Green">E67</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_1.html#E3:80"><i><font color="Green">E62</font></i></a>, <a class="txt" href="sprect_1.html#E6:80"><i><font color="Green">E65</font></i></a>, <a class="ref" href="sprect_1.html#T62" target="_self">Th62</a></i>;<br/>
<a NAME="E9:80"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T19">COMPTS_1:19</a></i>;<br/>
<i><font color="Green">E69</font></i>:  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sprect_1.html#T54" target="_self">Th54</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_1.html#E2:80"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sprect_1.html#E3:80"><i><font color="Green">E62</font></i></a>, <a class="txt" href="sprect_1.html#E4:80"><i><font color="Green">E63</font></i></a>, <a class="txt" href="sprect_1.html#E7:80"><i><font color="Green">E66</font></i></a>, <a class="ref" href="sprect_1.html#T62" target="_self">Th62</a></i>
<br/>.= 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="sprect_1.html#E2:80"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
;<br/>
<a NAME="E11:80"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T19">COMPTS_1:19</a></i>;<br/>
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sprect_1.html#T54" target="_self">Th54</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_1.html#E2:80"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sprect_1.html#E5:80"><i><font color="Green">E64</font></i></a>, <a class="txt" href="sprect_1.html#E6:80"><i><font color="Green">E65</font></i></a>, <a class="txt" href="sprect_1.html#E8:80"><i><font color="Green">E67</font></i></a>, <a class="ref" href="sprect_1.html#T62" target="_self">Th62</a></i>
<br/>.= 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font>

;<br/>
<b>hence </b> <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_1.html#E1:80"><i><font color="Green">E60</font></i></a>, <a class="txt" href="sprect_1.html#E9:80"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sprect_1.html#E10:80"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sprect_1.html#E11:80"><i><font color="Green">E70</font></i></a>, <a class="ref" href="sprect_1.html#T54" target="_self">Th54</a></i>
<br/>.= 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font>

;<br/><br/>


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