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<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>let </b><font color="Maroon">X'</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:53"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:53"/><b>then </b>
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<b>hence </b><a NAME="E3:53"/>
<font color="Maroon">X'</font> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D1">SEQ_4:def 1</a></i>;<br/>

<a NAME="E4:53"/>
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<b>hence </b><a NAME="E5:53"/>
<font color="Maroon">X'</font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>


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