<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">FX</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PM</font>, <font color="Maroon">GX</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">PM</font>;<br/>

<b>assume </b><a NAME="E1:30_1_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> holds <br/> ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> iff for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:30_1_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> holds <br/> ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> iff for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> holds <br/> ex <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<font color="Olive">p</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font> ) )
 ;<br/>

<a NAME="E3:30_1_2"/>
for <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">PM</font> holds <br/> ( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> iff <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><b>proof </b></a><div class="add">

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

<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_2_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 ;<br/><a NAME="E2:30_1_2_1_1"/><b>then </b>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:30_1_2_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_2_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 ;<br/><a NAME="E2:30_1_2_1_2"/><b>then </b>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:30_1_2_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:30_1_2_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> iff <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_1_2"/>
<font color="Maroon">FX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">GX</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T44">SETFAM_1:44</a></i>;<br/>


</div>
