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

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<a NAME="E1:35"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">j</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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:35"/>
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span>, <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="t_0topsp.html#R1">are_homeomorphic</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">ci</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">cj</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">cij</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span>,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">fi</font>, <font color="Olive">fj</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">fj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">fi</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">fj</font> );<br/>
<a NAME="E1:35_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span><br/> for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span>  ex <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 <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span>;<br/>



<a NAME="E1:35_1_1"/>
( <font color="Maroon">x</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">i</font> &amp; <font color="Maroon">y</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">j</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">fi</font> = <font color="Maroon">x</font>, <font color="Maroon">fj</font> = <font color="Maroon">y</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E3:35_1_1"/>
<font color="Maroon">fi</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fj</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T127">FINSEQ_2:127</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">fi</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fj</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>thus </b><a NAME="E5:35_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <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 <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span><br/> for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<a NAME="E4:35_1"/><i><font color="Green">E49</font></i>: 
<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="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:35_1"/><i><font color="Green">E51</font></i>: 
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="euclid.html#K14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E6:35_1"/><i><font color="Green">E52</font></i>: 
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> <span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</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> <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a><br/>

<a NAME="E8:35_1"/>
 <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> <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:35_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <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> <span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a><br/>

<b>thus </b><a NAME="E10:35_1"/><i><font color="Green">E60</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></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><br/>

<b>let </b><font color="Maroon">y</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="E2:35_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">y</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:35_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">h</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:35_1_2"/><i><font color="Green">E62</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 <b>and </b><br/><a NAME="E7:35_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T26">FINSEQ_2:26</a></i>;<br/>
<a NAME="E8:35_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">g</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="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">h</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="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:35_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
 ex <font color="Olive">fi</font>, <font color="Olive">fj</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Olive">fj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">fi</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">fj</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E10:35_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span> <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 <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:35_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:35_1_2"/>
<font color="Maroon">y</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/>


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

<b>thus </b><a NAME="E11:35_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:35_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x1b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1_3"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x1a</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="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">x1b</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="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:35_1_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1a</font>,<font color="Maroon">x1b</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>assume </b><a NAME="E5:35_1_3"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x2a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:35_1_3"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">x2a</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="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">x2b</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="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span> )
 <b>and </b><br/><a NAME="E8:35_1_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2a</font>,<font color="Maroon">x2b</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>consider </b><font color="Maroon">fi1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">fj1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E10:35_1_3"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">fi1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1a</font> &amp; <font color="Maroon">fj1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1b</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1a</font>,<font color="Maroon">x1b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fj1</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">fi2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">fj2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E12:35_1_3"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">fi2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2a</font> &amp; <font color="Maroon">fj2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2b</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x2a</font>,<font color="Maroon">x2b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fj2</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E13:35_1_3"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fi1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fj1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E14:35_1_3"/><b>then </b><i><font color="Green">E115</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fi1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fi2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fj1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fj2</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>assume </b><a NAME="E15:35_1_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E16:35_1_3"/><b>then </b>
( <font color="Maroon">fi1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi2</font> &amp; <font color="Maroon">fj1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fj2</font> )
 
<b>by </b><i>, , , , , </i>;<br/>
<b>hence </b><a NAME="E17:35_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , , </i>;<br/>


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

<a NAME="E12:35_1"/>
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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>  st <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> 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="35_1_4"><b>proof </b></a><div class="add">

<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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:35_1_4"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:35_1_4"/><b>then </b>
<font color="Maroon">P</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="euclid.html#K14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E3:35_1_4"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:35_1_4"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E5:35_1_4"/>
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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span>  st <font color="Olive">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> 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">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_4_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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:35_1_4_1"/><i><font color="Green">E119</font></i>: 
<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>
 ;<br/>

<a NAME="E2:35_1_4_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:35_1_4_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:35_1_4_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:35_1_4_1"/><i><font color="Green">E121</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">fx</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</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">r1</font> = <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2;<br/>

<b>thus </b><a NAME="E8:35_1_4_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</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="E9:35_1_4_1"/><i><font color="Green">E122</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">r1</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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">b1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>, <font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>, <font color="Maroon">b2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span><b> such that </b><br/><a NAME="E12:35_1_4_1"/><i><font color="Green">E123</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">b1</font>,<font color="Maroon">b2</font></span><a href="topreal7.html#K2" target="_self">]</a></span> &amp; <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">x1</font>,<font color="Maroon">x2</font></span><a href="topreal7.html#K2" target="_self">]</a></span> )
 <b>and </b><br/><a NAME="E13:35_1_4_1"/>
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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></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 <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b1</font>,<font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1">distance</a> of <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b2</font>,<font color="Maroon">x2</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">b1f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">b2f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E15:35_1_4_1"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">b1f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">b2f</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x1f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x2f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E17:35_1_4_1"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">x1f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x2f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">x2f</font> )
 <b>by </b><i/>;<br/>
<a NAME="E18:35_1_4_1"/><i><font color="Green">E127</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="topreal7.html#K2" target="_self">]</a></span>,<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"><a href="metric_1.html#K4">dist</a> <font color="Maroon">b1</font>,<font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">b2</font>,<font color="Maroon">x2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E19:35_1_4_1"/><i><font color="Green">E129</font></i>: 
 <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">r1</font>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E20:35_1_4_1"/>
<font color="Maroon">bb</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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E21:35_1_4_1"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">bb</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="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E22:35_1_4_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">bb</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="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fb</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fbb</font> = <font color="Maroon">fb</font>, <font color="Maroon">fxx</font> = <font color="Maroon">fx</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">b1i</font> = <font color="Maroon">b1</font>, <font color="Maroon">x1i</font> = <font color="Maroon">x1</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">i</font> ;<br/>
<b>reconsider </b><font color="Maroon">x2i</font> = <font color="Maroon">x2</font>, <font color="Maroon">b2i</font> = <font color="Maroon">b2</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">j</font> ;<br/>
<a NAME="E27:35_1_4_1"/>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">b1</font>,<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">b1</font>,<font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">b2</font>,<font color="Maroon">x2</font></span>)</span> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">b2</font>,<font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">b1</font>,<font color="Maroon">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">b2</font>,<font color="Maroon">x2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E28:35_1_4_1"/><b>then </b>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">b1</font>,<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">b2</font>,<font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> )
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E29:35_1_4_1"/><b>then </b><i><font color="Green">E132</font></i>: 
( <span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">i</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b1i</font>,<font color="Maroon">x1i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> &amp; <span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">j</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b2i</font>,<font color="Maroon">x2i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> )
 
;<br/>
<a NAME="E30:35_1_4_1"/><i><font color="Green">E133</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x1f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b2f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x2f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="ref" href="topreal7.html#T1" target="_self">Th1</a>, </i>;<br/>
<i><font color="Green">E136</font></i>: <span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">fbb</font>,<font color="Maroon">fxx</font> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fbb</font> <a href="euclid.html#K8">-</a> <font color="Maroon">fxx</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p4">(<span class="default"><font color="Maroon">fbb</font> <a href="euclid.html#K8">-</a> <font color="Maroon">fxx</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T29">EUCLID:29</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p5">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p5">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="topreal7.html#T1" target="_self">Th1</a>, , </i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p5">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p5">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p5">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p5">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p5">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T29">EUCLID:29</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T29">EUCLID:29</a></i>
;<br/>
<a NAME="E32:35_1_4_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116">RVSUM_1:116</a></i>;<br/>
<a NAME="E33:35_1_4_1"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal6.html#T6">TOPREAL6:6</a></i>;<br/>
<a NAME="E34:35_1_4_1"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E35:35_1_4_1"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
;<br/>
<a NAME="E36:35_1_4_1"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">i</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b1i</font>,<font color="Maroon">x1i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>;<br/>
<a NAME="E37:35_1_4_1"/><b>then </b><i><font color="Green">E137</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b1i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">b2i</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">i</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b1i</font>,<font color="Maroon">x1i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">j</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b2i</font>,<font color="Maroon">x2i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>;<br/>
<a NAME="E38:35_1_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">i</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b1i</font>,<font color="Maroon">x1i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">j</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">b2i</font>,<font color="Maroon">x2i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E39:35_1_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">fbb</font>,<font color="Maroon">fxx</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="topreal7.html#T3" target="_self">Th3</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E40:35_1_4_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">fb</font>,<font color="Maroon">fx</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
;<br/>
<a NAME="E41:35_1_4_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">fx</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="E42:35_1_4_1"/>
<font color="Maroon">b</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="topreal7.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:35_1_4"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:35_1_4"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</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> <span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:35_1"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>

<a NAME="E14:35_1"/>
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> <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>  st <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <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="35_1_5"><b>proof </b></a><div class="add">

<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> <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:35_1_5"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:35_1_5"/><b>then </b>
<font color="Maroon">P</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> <span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E3:35_1_5"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:35_1_5"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E5:35_1_5"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</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">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_5_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="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:35_1_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</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="E3:35_1_5_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">a</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="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" target="_self">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:35_1_5_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E5:35_1_5_1"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="topmetr.html#T16">TOPMETR:16</a></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="E8:35_1_5_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E9:35_1_5_1"/><i><font color="Green">E146</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">a</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</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>
;<br/>

<b>thus </b><a NAME="E10:35_1_5_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><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="E11:35_1_5_1"/><i><font color="Green">E147</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="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">bb2</font> = <font color="Maroon">bb</font>, <font color="Maroon">xx2</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> ;<br/>
<a NAME="E14:35_1_5_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="E15:35_1_5_1"/><b>then </b><i><font color="Green">E148</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></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/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">bb</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> ;<br/>
<a NAME="E17:35_1_5_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">g</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">h</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E19:35_1_5_1"/><i><font color="Green">E149</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E20:35_1_5_1"/><i><font color="Green">E179</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E21:35_1_5_1"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T26">FINSEQ_2:26</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:35_1_5_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">p</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="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E24:35_1_5_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">q</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="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span>
 <b>and </b><br/><a NAME="E25:35_1_5_1"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</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">p</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">fi</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">fj</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E29:35_1_5_1"/><i><font color="Green">E186</font></i>: 
( <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">fj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>and </b><br/><a NAME="E30:35_1_5_1"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">fj</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">gg</font> = <font color="Maroon">g</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">i</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">hh</font> = <font color="Maroon">h</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">j</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">h</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E34:35_1_5_1"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hh</font> )
 <b>and </b><br/><a NAME="E35:35_1_5_1"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">gg</font>,<font color="Maroon">hh</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">gg2</font> = <font color="Maroon">gg</font>, <font color="Maroon">a12</font> = <font color="Maroon">a</font> <a href="topreal7.html#K3" target="_self">`1</a>  as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">i</font> ;<br/>
<b>reconsider </b><font color="Maroon">hh2</font> = <font color="Maroon">hh</font>, <font color="Maroon">a22</font> = <font color="Maroon">a</font> <a href="topreal7.html#K4" target="_self">`2</a>  as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">j</font> ;<br/>
<a NAME="E38:35_1_5_1"/><i><font color="Green">E196</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="txt" href="topreal7.html#E1:35"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E39:35_1_5_1"/>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">bb2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">xx2</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <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="euclid.html#D6">EUCLID:def 6</a></i>;<br/>
<a NAME="E40:35_1_5_1"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">bb2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">xx2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
;<br/>
<a NAME="E41:35_1_5_1"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p4">(<span class="default"><font color="Maroon">bb2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">xx2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T29">EUCLID:29</a></i>;<br/>
<a NAME="E42:35_1_5_1"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fi</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="topreal7.html#T2" target="_self">Th2</a>, <a class="ref" href="topreal7.html#T3" target="_self">Th3</a>, <a class="ref" href="topreal7.html#T5" target="_self">Th5</a>, <a class="txt" href="topreal7.html#E1:35"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topreal7.html#E3:35_1"><i><font color="Green">E48</font></i></a>, </i>;<br/>
<a NAME="E43:35_1_5_1"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fi</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E44:35_1_5_1"/><b>then </b><i><font color="Green">E197</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>;<br/>
<a NAME="E45:35_1_5_1"/><i><font color="Green">E198</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">gg</font>,<font color="Maroon">hh</font></span><a href="topreal7.html#K2" target="_self">]</a></span>,<span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="topreal7.html#K3" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="topreal7.html#K4" target="_self">`2</a> </span>)</span></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"><a href="metric_1.html#K4">dist</a> <font color="Maroon">gg</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="topreal7.html#K3" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">hh</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="topreal7.html#K4" target="_self">`2</a> </span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E46:35_1_5_1"/><i><font color="Green">E199</font></i>: 
( <font color="Maroon">a12</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">a22</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="topreal7.html#T3" target="_self">Th3</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E47:35_1_5_1"/><i><font color="Green">E200</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p3">(<span class="default"><font color="Maroon">gg2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a12</font></span>)</span></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p3">(<span class="default"><font color="Maroon">hh2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a22</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116">RVSUM_1:116</a></i>;<br/>
<a NAME="E48:35_1_5_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fi</font></span>)</span></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fj</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116">RVSUM_1:116</a></i>;<br/>
<a NAME="E49:35_1_5_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p4">(<span class="default"><font color="Maroon">gg2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a12</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fj</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p4">(<span class="default"><font color="Maroon">hh2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a22</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fj</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="topreal7.html#E1:35"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topreal7.html#E5:35_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E50:35_1_5_1"/><b>then </b>
(  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p4">(<span class="default"><font color="Maroon">gg2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a12</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p4">(<span class="default"><font color="Maroon">hh2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a22</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E6:35_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<a NAME="E51:35_1_5_1"/><b>then </b>
(  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p4">(<span class="default"><font color="Maroon">gg2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a12</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp;  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10">abs</a> <span class="p4">(<span class="default"><font color="Maroon">hh2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a22</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E4:35_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E52:35_1_5_1"/><b>then </b>
(  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">gg2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a12</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp;  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">hh2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a22</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T29">EUCLID:29</a></i>;<br/>
<a NAME="E53:35_1_5_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">gg2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a12</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">hh2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a22</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
;<br/>
<a NAME="E54:35_1_5_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">i</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">gg2</font>,<font color="Maroon">a12</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">j</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">hh2</font>,<font color="Maroon">a22</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>;<br/>
<a NAME="E55:35_1_5_1"/><b>then </b><i><font color="Green">E201</font></i>: 
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">gg</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="topreal7.html#K3" target="_self">`1</a> </span>)</span> <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">hh</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="topreal7.html#K4" target="_self">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
;<br/>
<a NAME="E56:35_1_5_1"/>
(  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">gg</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="topreal7.html#K3" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">hh</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="topreal7.html#K4" target="_self">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">gg</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="topreal7.html#K3" target="_self">`1</a> </span>)</span> or  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">gg</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="topreal7.html#K3" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">hh</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="topreal7.html#K4" target="_self">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">hh</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="topreal7.html#K4" target="_self">`2</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/>
<a NAME="E57:35_1_5_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <span class="p1"><a href="topreal7.html#K2" target="_self">[</a><span class="default"><font color="Maroon">gg</font>,<font color="Maroon">hh</font></span><a href="topreal7.html#K2" target="_self">]</a></span>,<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="topreal7.html#T6" target="_self">Th6</a>, <a class="ref" href="topreal7.html#T7" target="_self">Th7</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E58:35_1_5_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">a</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E1:35"><i><font color="Green">E46</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E59:35_1_5_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E60:35_1_5_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="topreal7.html#T2" target="_self">Th2</a>, <a class="txt" href="topreal7.html#E1:35"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topreal7.html#E1:35_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:35_1_5"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<a NAME="E7:35_1_5"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</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="euclid.html#K14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<b>hence </b><a NAME="E8:35_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</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="euclid.html#K14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="tops_2.html#T67">TOPS_2:67</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:35_1"/>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:35"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">j</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>, <a href="euclid.html#K15">TOP-REAL</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="t_0topsp.html#R1">are_homeomorphic</a> 
 <b>by </b><i/>;<br/>


</div>
