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<b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>



<b>thus </b><a NAME="E1:29"/>
( ( for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff <font color="Olive">r</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> ) ) implies <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1"/>
for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff <font color="Olive">r</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 ;<br/>

<a NAME="E2:29_1"/><b>then </b>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 
;<br/>
<b>hence </b><a NAME="E3:29_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:29"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>let </b><font color="Maroon">r</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/>

<b>thus </b><a NAME="E3:29"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> implies <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 ;<br/>

<b>thus </b><a NAME="E4:29"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> implies <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/>


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