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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L2">RLSStruct</a> ;<br/><b>let </b><font color="Maroon">F</font> be  <a href="rltopsp1.html#V4" target="_self">circled-membered</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D6" target="_self">RLTOPSP1:def 6</a><br/>





<b>assume </b><a NAME="E1:55"/><i><font color="Green">E36</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:55"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:55"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E5:55"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 ;<br/>
<a NAME="E6:55"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E4:55"><i><font color="Green">E37</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:55_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:55"><i><font color="Green">E37</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="rltopsp1.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E4:55_1"/>
<font color="Maroon">Y'</font> is <a href="rltopsp1.html#V3" target="_self">circled</a>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E5:55_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">Y'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:55_1"/>
<font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">Y'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:55_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:55"><i><font color="Green">E36</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:55"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E5:55"><i><font color="Green">E38</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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