<?xml version="1.0"?>
<div class="add">

<b>let </b><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">N</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/>



<b>set </b><font color="Maroon">S</font> =  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font>;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font>;<br/>
<a NAME="E1:33"/><i><font color="Green">E46</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> : <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">X2</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <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>, <font color="Olive">X2</font> is   <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">N</font></span>)</span> : ( <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> &amp; <font color="Olive">X2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span> ) </span>}</span>  </span>}</span> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E2:33"/><i><font color="Green">E47</font></i>: 
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3">Family_open_set</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D6">PCOMPS_1:def 6</a></i>;<br/>
<a NAME="E3:33"/><i><font color="Green">E48</font></i>: 
 <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/>
<a NAME="E4:33"/><i><font color="Green">E49</font></i>: 
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</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">N</font>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">N</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D6">PCOMPS_1:def 6</a></i>;<br/>
<i><font color="Green">E51</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span></span>)</span>
<b>by </b><i>, , , </i>
;<br/>
<a NAME="E6:33"/>
the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:33_2"/>
the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_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="E1:33_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:33_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:33_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">X2</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <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>, <font color="Olive">X2</font> is   <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">N</font></span>)</span> : ( <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> &amp; <font color="Olive">X2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span> ) </span>}</span> 
 <b>by </b><i/>;<br/>
<a NAME="E5:33_2_1"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:33_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_2_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E4:33_2_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:33_2_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">X2</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <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>, <font color="Olive">X2</font> is   <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">N</font></span>)</span> : ( <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> &amp; <font color="Olive">X2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">X1</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>, <font color="Maroon">X2</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">N</font></span>)</span><b> such that </b><br/><a NAME="E7:33_2_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E8:33_2_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>
 <b>and </b><br/><a NAME="E9:33_2_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span>
 ;<br/>
<b>consider </b><font color="Maroon">z1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:33_2_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>and </b><br/><a NAME="E12:33_2_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>and </b><br/><a NAME="E13:33_2_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">z1</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">z2</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E17:33_2_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E18:33_2_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z1</font>,<font color="Maroon">r1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font>
 <b>by </b><i>, , , <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E20:33_2_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E21:33_2_1_1"/><i><font color="Green">E115</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z2</font>,<font color="Maroon">r2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font>
 <b>by </b><i>, , , <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<b>take </b><font color="Maroon">r</font> =  <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>;<br/>

<b>thus </b><a NAME="E22:33_2_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="topreal7.html#T1" target="_self">Th1</a>, , <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>

<b>let </b><font color="Maroon">b</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="E23:33_2_1_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">y1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>, <font color="Maroon">y2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E26:33_2_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">bb</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="topreal7.html#K2" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E27:33_2_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="topreal7.html#K2" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E28:33_2_1_1"/><i><font color="Green">E120</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">bb</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E29:33_2_1_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 
<b>by </b><i>, <a class="ref" href="topreal7.html#T3" target="_self">Th3</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E30:33_2_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">bb</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E31:33_2_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">bb</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
;<br/>
<a NAME="E32:33_2_1_1"/><i><font color="Green">E123</font></i>: 
(  <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> &amp;  <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E33:33_2_1_1"/>
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal7.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E34:33_2_1_1"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="txt" href="topreal7.html#E1:33"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:33_2_1_1"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E2:33"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E36:33_2_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 
;<br/>
<a NAME="E37:33_2_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z1</font>,<font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E38:33_2_1_1"/>
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal7.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E39:33_2_1_1"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="txt" href="topreal7.html#E1:33"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E40:33_2_1_1"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E2:33"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E41:33_2_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 
;<br/>
<a NAME="E42:33_2_1_1"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z2</font>,<font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E43:33_2_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="topreal7.html#T2" target="_self">Th2</a>, , <a class="txt" href="topreal7.html#E3:33"><i><font color="Green">E48</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E44:33_2_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:33_2_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>


</div><b>end;</b></div>

<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="E2:33_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i/>;<br/>
<a NAME="E4:33_2"/><i><font color="Green">E127</font></i>: 
 <a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <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>, <font color="Olive">Y1</font> is   <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">N</font></span>)</span> : ( <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D6">BORSUK_1:def 6</a></i>;<br/>
<a NAME="E5:33_2"/><i><font color="Green">E129</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:33_2_2"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">Y</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T53">BORSUK_1:53</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">u</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="E2:33_2_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">uu</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:33_2_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E6:33_2_2"/><i><font color="Green">E133</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">uu</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , , <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<a NAME="E7:33_2_2"/>
<font color="Maroon">uu</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span>
 
;<br/>
<a NAME="E8:33_2_2"/><b>then </b>
<font color="Maroon">uu</font> <a href="hidden.html#R2">in</a> <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">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">u1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">u2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:33_2_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E11:33_2_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E12:33_2_2"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">u2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u1</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">u2</font> = <font color="Maroon">u2</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u1</font>,<font color="Maroon">r</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> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B2</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u2</font>,<font color="Maroon">r</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">N</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E17:33_2_2"/>
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u1</font>,<font color="Maroon">r</font> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u2</font>,<font color="Maroon">r</font> )
 
<b>by </b><i>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/>
<a NAME="E18:33_2_2"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">B2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E19:33_2_2"/><i><font color="Green">E142</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">B2</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_2_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="E1:33_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">B2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_2_2_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>and </b><br/><a NAME="E4:33_2_2_1"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 <b>and </b><br/><a NAME="E5:33_2_2_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="topreal7.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">q1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>, <font color="Maroon">q2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E9:33_2_2_1"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">uu</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">q1</font></span><a href="topreal7.html#K2" target="_self">]</a></span> &amp; <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="topreal7.html#K2" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">q2</font></span><a href="topreal7.html#K2" target="_self">]</a></span> )
 <b>and </b><br/><a NAME="E10:33_2_2_1"/><i><font color="Green">E147</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">uu</font>,<span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="topreal7.html#K2" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E11:33_2_2_1"/>
( <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 
<b>by </b><i>, <a class="ref" href="topreal7.html#T2" target="_self">Th2</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E12:33_2_2_1"/><b>then </b>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="topreal7.html#T1" target="_self">Th1</a>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E13:33_2_2_1"/><b>then </b>
( the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
;<br/>
<a NAME="E14:33_2_2_1"/><b>then </b>
 <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">N</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/>
<a NAME="E15:33_2_2_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">uu</font>,<span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="topreal7.html#K2" target="_self">]</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E16:33_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">uu</font>,<font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E17:33_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E20:33_2_2"/>
( <font color="Maroon">B1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">B2</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/>
<a NAME="E21:33_2_2"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">B2</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E22:33_2_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:33_2"/>
 <a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <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>, <font color="Olive">Y1</font> is   <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">N</font></span>)</span> : ( <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> &amp; <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</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="E1:33_2_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X1</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>, <font color="Maroon">Y1</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">N</font></span>)</span><b> such that </b><br/><a NAME="E3:33_2_3"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E4:33_2_3"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:33_2_3"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i/>;<br/>
<a NAME="E6:33_2_3"/>
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> &amp; <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:33_2_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <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>, <font color="Olive">Y1</font> is   <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">N</font></span>)</span> : ( <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> &amp; <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:33_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:33"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <font color="Maroon">M</font>,<font color="Maroon">N</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
