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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>,  <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K3">R^1</a> ] <b>means </b>for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><br/> for <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K3">R^1</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> holds <br/><font color="Olive">t</font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Olive">s</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span>;<br/>
<a NAME="E1:29_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K3">R^1</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>

<b>consider </b><font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:29_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span>
 ;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>thus </b><a NAME="E4:29_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>,<font color="Maroon">t</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E3:29_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E4:29_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Olive">y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:29_1_1"/>
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">M</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D6">PCOMPS_1:def 6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E7:29_1_1"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> holds  <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Olive">x</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
