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<b>let </b><font color="Maroon">D</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="E1:99"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">D</font> is <a href="jordan2c.html#V1">Bounded</a>
 ;<br/>

<a NAME="E2:99"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:99"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:99"/><i><font color="Green">E58</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</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">D</font></span>)</span> &amp;  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span> <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> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T48">SPRECT_1:48</a></i>;<br/>
<a NAME="E5:99"/><i><font color="Green">E59</font></i>: 
<a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="topreal6.html#E1"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="sprect_1.html#T46">SPRECT_1:46</a></i>;<br/>
<a NAME="E6:99"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E7:99"/><b>then </b>
<a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E8:99"/><b>then </b>
<a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="topreal6.html#E1:99"><i><font color="Green">E56</font></i></a>, <a class="ref" href="rcomp_1.html#T3">RCOMP_1:3</a></i>;<br/>
<b>then </b> <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">D</font></span>)</span> = 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6">Cl</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">D</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <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> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E10:99"/>
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal6.html#T6" target="_self">Th6</a></i>;<br/>


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