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<b>let </b><font color="Maroon" title="c1">D</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">D</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> implies  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">N-bound</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">N-bound</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">D</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:101"/><i><font color="Green" title="E49">A1</font></i>: 
<font color="Maroon" title="c1">D</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">N-bound</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">N-bound</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">D</font></span>)</span></span><br/>

<a NAME="E2:101"/><b>then </b><i><font color="Green" title="E50">A2</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">D</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T71" target="_self" title="TOPREAL6:th.71">Th71</a>, <a class="ref" href="topreal6.html#T88" target="_self" title="TOPREAL6:th.88">Th88</a></i>;<br/>
<a NAME="E3:101"/><i><font color="Green" title="E51">A3</font></i>: 
(  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">N-bound</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">D</font></span>)</span> &amp;  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">N-bound</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T50" title="SPRECT_1:th.50">SPRECT_1:50</a></i>;<br/>
<a NAME="E4:101"/><i><font color="Green" title="E52">A4</font></i>: 
<a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">D</font></span>)</span> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>
 
<b>by </b><i><a class="txt" href="#E2:101"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="sprect_1.html#T47" title="SPRECT_1:th.47">SPRECT_1:47</a></i>;<br/>
<a NAME="E5:101"/>
<font color="Maroon" title="c1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E6:101"/><b>then </b>
<a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">D</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>
 
<b>by </b><i><a class="txt" href="#E4:101"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="ref" href="xxreal_2.html#T43" title="XXREAL_2:th.43">XXREAL_2:43</a></i>;<br/>
<b>then </b> <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">D</font></span>)</span> = 
 <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal6.html#T78" target="_self" title="TOPREAL6:th.78">Th78</a></i>
<br/>.= 
 <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:101"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="topreal6.html#T93" target="_self" title="TOPREAL6:th.93">Th93</a></i>
;<br/>
<b>hence </b><a NAME="E8:101"/>
 <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">N-bound</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">N-bound</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:101"><i><font color="Green" title="E51">A3</font></i></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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