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<b>let </b><font color="Maroon">C1</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>let </b><font color="Maroon">C2</font> be  non <a href="xboole_0.html#V1">empty</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>, <font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</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>assume </b><a NAME="E1:9"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> &amp; not <a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">C2</font> is <a href="xboole_0.html#V1">empty</a> &amp; <a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">C2</font> is <a href="seq_4.html#V2">bounded_below</a> )
 ;<br/>

<b>set </b><font color="Maroon">P1</font> = <a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">C1</font>;<br/>
<b>set </b><font color="Maroon">P2</font> = <a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">C2</font>;<br/>
<b>set </b><font color="Maroon">PS</font> = <a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font>;<br/>
<a NAME="E2:9"/><i><font color="Green">E51</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">C1</font></span>)</span> &amp;  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">C2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T48">SPRECT_1:48</a></i>;<br/>
<a NAME="E3:9"/><i><font color="Green">E52</font></i>: 
( not <a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">C1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">C1</font> is <a href="seq_4.html#V2">bounded_below</a> &amp; not <a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">C2</font> is <a href="xboole_0.html#V1">empty</a> &amp; <a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">C2</font> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i>, <a class="ref" href="sprect_1.html#T46">SPRECT_1:46</a></i>;<br/>
<b>thus </b> <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">S</font> = 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_1.html#T48">SPRECT_1:48</a></i>
<br/>.= 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">C1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">C2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T153">RELAT_1:153</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">C1</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="sprect_1.html#T52">SPRECT_1:52</a></i>
;<br/>


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