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<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:37_1_2"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="bhsp_2.html#K4">Ball</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:37_1_2"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="bhsp_2.html#K4">Ball</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<a NAME="E3:37_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:37_1_2_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="bhsp_2.html#K4">Ball</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:37_1_2"><i><font color="Green">E21</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_1_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rusub_5.html#E2:37_1_2"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:37_1_2_1_2"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="bhsp_2.html#K4">Ball</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_5.html#E2:37_1_2"><i><font color="Green">E22</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_1_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:37_1_2"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:37_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:37_1_2_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:37_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T44">SETFAM_1:44</a></i>;<br/>


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