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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:35_1_2"/><i><font color="Green">E25</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 <b>and </b><br/><a NAME="E2:35_1_2"/><i><font color="Green">E26</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="topgen_2.html#M1">Neighborhood_System</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) )
 <b>and </b><br/><a NAME="E3:35_1_2"/><i><font color="Green">E27</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 <b>and </b><br/><a NAME="E4:35_1_2"/><i><font color="Green">E28</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="topgen_2.html#M1">Neighborhood_System</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="topgen_2.html#M1">Neighborhood_System</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:35_1_2"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>and </b><br/><a NAME="E7:35_1_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E2:35_1_2"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="topgen_2.html#M1">Neighborhood_System</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:35_1_2"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>and </b><br/><a NAME="E10:35_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:35_1_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="struct_0.html#M1">Element</a> of  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  ;<br/><a NAME="E3:35_1_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E4:35_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="topgen_5.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E5:35_1_2_1"/><b>then </b>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) or ( <font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a> </span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:35_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="topgen_5.html#E6:35_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="topgen_5.html#E9:35_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="topgen_5.html#E7:35_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topgen_5.html#E10:35_1_2"><i><font color="Green">E32</font></i></a></i>;<br/><b>hence </b><a NAME="E6:35_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E12:35_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E1:35_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="topgen_5.html#E3:35_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>hence </b><a NAME="E13:35_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="topgen_5.html#T28" target="_self">Th28</a></i>;<br/>


</div>
