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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">pmet</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">pmet</font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">pmet'</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>  st <font color="Maroon">pmet</font> <a href="funct_2.html#R1">=</a> <font color="Olive">pmet'</font> holds <br/><font color="Olive">pmet'</font> is <a href="pscomp_1.html#V9">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>



<b>set </b><font color="Maroon">rn</font> = 0;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>, <font color="Maroon">infA</font> = <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</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>reconsider </b><font color="Maroon">InfA</font> =  <a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<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>assume </b><a NAME="E5:21"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E6:21"/>
<font color="Maroon">infA</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">infa</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="E1:21_1"/>
<font color="Maroon">infa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">infA</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">infa</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:21_1"/>
<font color="Maroon">infa</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:21_1"/>
<font color="Maroon">infa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:21"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">infA</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E8:21"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds   <a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet</font>,<font color="Olive">p</font> is <a href="pscomp_1.html#V9">continuous</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:21"/><b>then </b>
(  <a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font> is <a href="pscomp_1.html#V9">continuous</a> &amp; 0 is   <a href="pre_topc.html#NM2">Point</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/>
<a NAME="E10:21"/><b>then </b>
( <font color="Maroon">InfA</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">Z</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T10">PCOMPS_1:10</a>, <a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<a NAME="E11:21"/><b>then </b>
( <font color="Maroon">InfA</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">InfA</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Z</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a>, <a class="ref" href="tops_2.html#T57">TOPS_2:57</a></i>;<br/>
<a NAME="E12:21"/><b>then </b>
( <font color="Maroon">InfA</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E13:21"/>
<span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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