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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-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="E1:112"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:112"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-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="E3:112"/><span class="kw">then </span>
 <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> 
 
<span class="kw">by</span> <span class="lab"><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></span>;<br/>
<a NAME="E4:112"/><span class="kw">then </span>
<a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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#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="E5:112"/><span class="kw">then </span>
<a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">D</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="txt" href="#E1:112"><span class="lab"><font color="Green" title="E49">A1</font></span></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#T44" title="XXREAL_2:th.44">XXREAL_2:44</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E51">A3</font></span>:  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">D</font></span>)</span> = 
 <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <span class="p1">(<span class="default"><a href="measure6.html#K6" title="MEASURE6:func.6">Cl</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T77" target="_self" title="TOPREAL6:th.77">Th77</a></span>
<br/>.= 
 <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="topreal6.html#T93" target="_self" title="TOPREAL6:th.93">Th93</a></span>
;<br/>
<a NAME="E7:112"/>
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">D</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#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T49" title="SPRECT_1:th.49">SPRECT_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:112"/>
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:112"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="sprect_1.html#T49" title="SPRECT_1:th.49">SPRECT_1:49</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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