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<b>let </b><font color="Maroon">N</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>, <font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <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">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>;<br/>



<a NAME="E1:4"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:4"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/> for <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font><br/> for <font color="Olive">u1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> holds <br/> ex <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">u</font>,<font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">u1</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> ) )
 ;<br/><a NAME="E2:4_1"/>
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/> for <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font><br/> for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">u</font>,<font color="Olive">r</font> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>;<br/>





<b>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> ;<br/>
<b>assume </b><a NAME="E2:4_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font> )
 ;<br/>

<a NAME="E3:4_1_1"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> holds <br/> ex <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>;<br/>

<b>assume </b><a NAME="E1:4_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:4_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">wf</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> ;<br/>
<a NAME="E4:4_1_1_1"/>
<font color="Maroon">P'</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T21">TOPMETR:21</a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E6:4_1_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">wf</font>,<font color="Maroon">r1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 <b>by </b><i>, , <a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">r1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E9:4_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E10:4_1_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Maroon">wf</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E12:4_1_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:4_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">s</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">wx</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<a NAME="E3:4_1_1_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">wx</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E4:4_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">wx</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">wfx</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">wx</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E6:4_1_1_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">wf</font>,<font color="Maroon">wfx</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:4_1_1_1_1"/><b>then </b>
<font color="Maroon">wfx</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">wf</font>,<font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E8:4_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:4_1_1_1"/>
 ex <font color="Olive">s2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">s2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Olive">s2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:4_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:4_1"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan2b.html#T20">JORDAN2B:20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:4"/>
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/> for <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font><br/> for <font color="Olive">u1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> holds <br/> ex <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">u</font>,<font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">u1</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> ) ) ) implies <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> )
 ;<br/>


</div>
