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<span class="kw">let </span><font color="Maroon" title="c1">C1</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C2</font>, <font color="Olive" title="b2">S</font> being 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>  st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">C2</font> &amp; not <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">C2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">C2</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  holds <br/> <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Olive" title="b1">C2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c3">S</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">C2</font> &amp; not <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">C2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">C2</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  implies  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c2">C2</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">and </span><br/><a NAME="E2:10"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
( not <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">C2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">C2</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c2">C2</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">P1</font> = <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">C1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">P2</font> = <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">C2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">PS</font> = <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">S</font>;<br/>
<a NAME="E3:10"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">C1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T48" title="SPRECT_1:th.48">SPRECT_1:48</a></span>;<br/>
<a NAME="E4:10"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
( not <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">C1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">C1</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T46" title="SPRECT_1:th.46">SPRECT_1:46</a></span>;<br/>
<a NAME="E5:10"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c2">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">C2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T48" title="SPRECT_1:th.48">SPRECT_1:48</a></span>;<br/>
<span class="kw">thus </span> <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c3">S</font> = 
 <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">S</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T48" title="SPRECT_1:th.48">SPRECT_1:48</a></span>
<br/>.= 
 <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">C1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">C2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></span>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c2">C2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="sprect_1.html#T52" title="SPRECT_1:th.52">SPRECT_1:52</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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