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<b>let </b><font color="Maroon" title="c1">S</font> be   <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font> being 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>  st <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">C2</font> holds <br/> <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Olive" title="b1">C1</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Olive" title="b2">C2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c3">C2</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">C2</font> implies  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c2">C1</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c3">C2</font></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:68"/><i><font color="Green" title="E50">A1</font></i>: 
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">C2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c2">C1</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c3">C2</font></span>)</span></span><br/>

<a NAME="E2:68"/><i><font color="Green" title="E51">A2</font></i>: 
(  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">C1</font></span>)</span> &amp;  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c3">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">C2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T50" target="_self" title="SPRECT_1:th.50">Th50</a></i>;<br/>
<a NAME="E3:68"/><i><font color="Green" title="E52">A3</font></i>: 
( not <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">C1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">C1</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> &amp; not <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">C2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">C2</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T47" target="_self" title="SPRECT_1:th.47">Th47</a></i>;<br/>
<b>thus </b> <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c1">S</font> = 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">S</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_1.html#T50" target="_self" title="SPRECT_1:th.50">Th50</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">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#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">C2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sprect_1.html#E1:68"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c2">C1</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c3">C2</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_1.html#E2:68"><i><font color="Green" title="E51">A2</font></i></a>, <a class="txt" href="sprect_1.html#E3:68"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="sprect_1.html#T53" target="_self" title="SPRECT_1:th.53">Th53</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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