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

<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<a NAME="E1:35"/>
 <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>, <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="t_0topsp.html#R1" title="T_0TOPSP:pred.1">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" title="c3">ci</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">cj</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">cij</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">fi</font>, <font color="Olive" title="b2">fj</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">fi</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">fj</font> );<br/>
<a NAME="E1:35_1"/><i><font color="Green" title="E20">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span>  ex <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">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" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span>;<br/>



<a NAME="E1:35_1_1"/>
( <font color="Maroon" title="c6">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c7">y</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">j</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">fi</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">fj</font> = <font color="Maroon" title="c7">y</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">u</font> = <font color="Maroon" title="c8">fi</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">fj</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T127" title="FINSEQ_2:th.127">FINSEQ_2:127</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">u</font>
;<br/>

<b>thus </b><a NAME="E4:35_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c10">u</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green" title="E21">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c6">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="topreal7.html#E1:35_1"><i><font color="Green" title="E20">A1</font></i></a>);<br/></i>
<a NAME="E4:35_1"/><i><font color="Green" title="E22">A3</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal7.html#D1" target="_self" title="TOPREAL7:def.1">Def1</a></i>;<br/>
<a NAME="E5:35_1"/><i><font color="Green" title="E23">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E6:35_1"/><i><font color="Green" title="E24">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="topreal7.html#E4:35_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="topreal7.html#E5:35_1"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a><br/>

<b>thus </b><a NAME="E8:35_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a><br/>

<b>thus </b><a NAME="E9:35_1"/><i><font color="Green" title="E25">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">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" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:35_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c8">y</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:35_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c11">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E6:35_1_2"/><i><font color="Green" title="E26">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> )
 <b>and </b><br/><a NAME="E7:35_1_2"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">h</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T26" title="FINSEQ_2:th.26">FINSEQ_2:26</a></i>;<br/>
<a NAME="E8:35_1_2"/><i><font color="Green" title="E28">A9</font></i>: 
( <font color="Maroon" title="c10">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span> &amp; <font color="Maroon" title="c11">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E6:35_1_2"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="topreal7.html#T17" target="_self" title="TOPREAL7:th.17">Th17</a></i>;<br/>
<a NAME="E9:35_1_2"/><b>then </b><i><font color="Green" title="E29">A10</font></i>: 
 ex <font color="Olive" title="b1">fi</font>, <font color="Olive" title="b2">fj</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">fi</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">g</font> &amp; <font color="Olive" title="b2">fj</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">h</font> &amp; <font color="Maroon" title="c7">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c11">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">fi</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">fj</font> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E3:35_1"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<a NAME="E10:35_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c11">h</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topreal7.html#E8:35_1_2"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:35_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c11">h</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:35_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="topreal7.html#E7:35_1_2"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="topreal7.html#E9:35_1_2"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E10:35_1"/><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c7">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">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" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:35_1_3"/>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x1a</font>, <font color="Maroon" title="c11">x1b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1_3"/><i><font color="Green" title="E27">A12</font></i>: 
( <font color="Maroon" title="c10">x1a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span> &amp; <font color="Maroon" title="c11">x1b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:35_1_3"/><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x1a</font>,<font color="Maroon" title="c11">x1b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="topreal7.html#E4:35_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="topreal7.html#E6:35_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>assume </b><a NAME="E5:35_1_3"/>
<font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">x2a</font>, <font color="Maroon" title="c13">x2b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:35_1_3"/><i><font color="Green" title="E29">A14</font></i>: 
( <font color="Maroon" title="c12">x2a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span> &amp; <font color="Maroon" title="c13">x2b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span> )
 <b>and </b><br/><a NAME="E8:35_1_3"/><i><font color="Green" title="E30">A15</font></i>: 
<font color="Maroon" title="c9">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x2a</font>,<font color="Maroon" title="c13">x2b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="topreal7.html#E4:35_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="topreal7.html#E6:35_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">fi1</font>, <font color="Maroon" title="c15">fj1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E10:35_1_3"/><i><font color="Green" title="E31">A16</font></i>: 
( <font color="Maroon" title="c14">fi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x1a</font> &amp; <font color="Maroon" title="c15">fj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1b</font> &amp; <font color="Maroon" title="c7">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x1a</font>,<font color="Maroon" title="c11">x1b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">fi1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c15">fj1</font> )
 <b>by </b><i><a class="txt" href="topreal7.html#E3:35_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="topreal7.html#E3:35_1_3"><i><font color="Green" title="E27">A12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">fi2</font>, <font color="Maroon" title="c17">fj2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E12:35_1_3"/><i><font color="Green" title="E32">A17</font></i>: 
( <font color="Maroon" title="c16">fi2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2a</font> &amp; <font color="Maroon" title="c17">fj2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2b</font> &amp; <font color="Maroon" title="c7">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">x2a</font>,<font color="Maroon" title="c13">x2b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">fi2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c17">fj2</font> )
 <b>by </b><i><a class="txt" href="topreal7.html#E3:35_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="topreal7.html#E7:35_1_3"><i><font color="Green" title="E29">A14</font></i></a></i>;<br/>
<a NAME="E13:35_1_3"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">fi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">fj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E3:35_1_3"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="topreal7.html#E10:35_1_3"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E14:35_1_3"/><b>then </b><i><font color="Green" title="E33">A18</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">fi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">fi2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">fj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">fj2</font> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E7:35_1_3"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="topreal7.html#E12:35_1_3"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<b>assume </b><a NAME="E15:35_1_3"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>
 ;<br/>

<a NAME="E16:35_1_3"/><b>then </b>
( <font color="Maroon" title="c14">fi1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c16">fi2</font> &amp; <font color="Maroon" title="c15">fj1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">fj2</font> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E4:35_1_3"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="topreal7.html#E8:35_1_3"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="topreal7.html#E10:35_1_3"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="topreal7.html#E12:35_1_3"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="topreal7.html#E14:35_1_3"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="topreal7.html#T6" target="_self" title="TOPREAL7:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E17:35_1_3"/>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font>
 <b>by </b><i><a class="txt" href="topreal7.html#E4:35_1_3"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="topreal7.html#E8:35_1_3"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="topreal7.html#E10:35_1_3"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="topreal7.html#E12:35_1_3"><i><font color="Green" title="E32">A17</font></i></a></i>;<br/>


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

<a NAME="E11:35_1"/><i><font color="Green" title="E27">A19</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E12:35_1"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>  st <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">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" title="c8">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:35_1_4"/>
<font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:35_1_4"/><b>then </b>
<font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E3:35_1_4"/><b>then </b><i><font color="Green" title="E28">A20</font></i>: 
<font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:35_1_4"/><i><font color="Green" title="E29">A21</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E5:35_1_4"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> holds <br/> ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">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" title="c9">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:35_1_4_1"/><i><font color="Green" title="E30">A22</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font>
 ;<br/>

<a NAME="E2:35_1_4_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">fx</font> = <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:35_1_4_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E1:35_1_4_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:35_1_4_1"/><i><font color="Green" title="E31">A23</font></i>: 
<font color="Maroon" title="c11">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E7:35_1_4_1"/><i><font color="Green" title="E32">A24</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c10">fx</font>,<font color="Maroon" title="c11">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P</font>
 <b>by </b><i><a class="txt" href="topreal7.html#E3:35_1_4"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></i>;<br/>
<b>take </b><font color="Maroon" title="c12">r1</font> = <font color="Maroon" title="c11">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2;<br/>

<b>thus </b><a NAME="E8:35_1_4_1"/>
<font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="topreal7.html#E6:35_1_4_1"><i><font color="Green" title="E31">A23</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>

<b>let </b><font color="Maroon" title="c13">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E9:35_1_4_1"/><i><font color="Green" title="E33">A25</font></i>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c12">r1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c14">bb</font> = <font color="Maroon" title="c13">b</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c15">b1</font>, <font color="Maroon" title="c16">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>, <font color="Maroon" title="c17">b2</font>, <font color="Maroon" title="c18">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span><b> such that </b><br/><a NAME="E12:35_1_4_1"/><i><font color="Green" title="E34">A26</font></i>: 
( <font color="Maroon" title="c14">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c17">b2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c18">x2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> )
 <b>and </b><br/><a NAME="E13:35_1_4_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c14">bb</font>,<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">x1</font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c17">b2</font>,<font color="Maroon" title="c18">x2</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal7.html#D1" target="_self" title="TOPREAL7:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">b1f</font>, <font color="Maroon" title="c20">b2f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E15:35_1_4_1"/><i><font color="Green" title="E35">A27</font></i>: 
( <font color="Maroon" title="c19">b1f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b1</font> &amp; <font color="Maroon" title="c20">b2f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b2</font> &amp; <font color="Maroon" title="c7">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c17">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b1f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c20">b2f</font> )
 <b>by </b><i><a class="txt" href="topreal7.html#E3:35_1"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">x1f</font>, <font color="Maroon" title="c22">x2f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E17:35_1_4_1"/><i><font color="Green" title="E36">A28</font></i>: 
( <font color="Maroon" title="c21">x1f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> &amp; <font color="Maroon" title="c22">x2f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x2</font> &amp; <font color="Maroon" title="c7">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c18">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x1f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c22">x2f</font> )
 <b>by </b><i><a class="txt" href="topreal7.html#E3:35_1"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<a NAME="E18:35_1_4_1"/><i><font color="Green" title="E37">A29</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c17">b2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span>,<span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c16">x1</font>,<font color="Maroon" title="c18">x2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c17">b2</font>,<font color="Maroon" title="c18">x2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal7.html#T20" target="_self" title="TOPREAL7:th.20">Th20</a></i>;<br/>
<a NAME="E19:35_1_4_1"/><i><font color="Green" title="E38">A30</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c14">bb</font>,<font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r1</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E9:35_1_4_1"><i><font color="Green" title="E33">A25</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E20:35_1_4_1"/>
<font color="Maroon" title="c14">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E21:35_1_4_1"/><b>then </b><i><font color="Green" title="E39">A31</font></i>: 
<font color="Maroon" title="c14">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E22:35_1_4_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c23">fb</font> = <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">b</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">fbb</font> = <font color="Maroon" title="c23">fb</font>, <font color="Maroon" title="c25">fxx</font> = <font color="Maroon" title="c10">fx</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c26">b1i</font> = <font color="Maroon" title="c15">b1</font>, <font color="Maroon" title="c27">x1i</font> = <font color="Maroon" title="c16">x1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">i</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c28">x2i</font> = <font color="Maroon" title="c18">x2</font>, <font color="Maroon" title="c29">b2i</font> = <font color="Maroon" title="c17">b2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">j</font> ;<br/>
<a NAME="E27:35_1_4_1"/>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c17">b2</font>,<font color="Maroon" title="c18">x2</font></span>)</span> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c17">b2</font>,<font color="Maroon" title="c18">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c17">b2</font>,<font color="Maroon" title="c18">x2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E28:35_1_4_1"/><b>then </b><i><font color="Green" title="E40">A32</font></i>: 
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c15">b1</font>,<font color="Maroon" title="c16">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r1</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c17">b2</font>,<font color="Maroon" title="c18">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r1</font> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E12:35_1_4_1"><i><font color="Green" title="E34">A26</font></i></a>, <a class="txt" href="topreal7.html#E18:35_1_4_1"><i><font color="Green" title="E37">A29</font></i></a>, <a class="txt" href="topreal7.html#E19:35_1_4_1"><i><font color="Green" title="E38">A30</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E29:35_1_4_1"/><i><font color="Green" title="E41">A33</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">b1f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">x1f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">b2f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">x2f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E15:35_1_4_1"><i><font color="Green" title="E35">A27</font></i></a>, <a class="txt" href="topreal7.html#E17:35_1_4_1"><i><font color="Green" title="E36">A28</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<i><font color="Green" title="E42">A34</font></i>: <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c24">fbb</font>,<font color="Maroon" title="c25">fxx</font> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">fbb</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">fxx</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">fbb</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">fxx</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></i>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c26">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c27">x1i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c29">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c28">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topreal7.html#E12:35_1_4_1"><i><font color="Green" title="E34">A26</font></i></a>, <a class="txt" href="topreal7.html#E15:35_1_4_1"><i><font color="Green" title="E35">A27</font></i></a>, <a class="txt" href="topreal7.html#E17:35_1_4_1"><i><font color="Green" title="E36">A28</font></i></a>, <a class="txt" href="topreal7.html#E29:35_1_4_1"><i><font color="Green" title="E41">A33</font></i></a>, <a class="ref" href="topreal7.html#T16" target="_self" title="TOPREAL7:th.16">Th16</a></i>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c26">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c27">x1i</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c29">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c28">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal7.html#T11" target="_self" title="TOPREAL7:th.11">Th11</a></i>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c26">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c27">x1i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c29">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c28">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c26">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c27">x1i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c28">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></i>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c27">x1i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c28">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></i>
;<br/>
<a NAME="E31:35_1_4_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c27">x1i</font></span>)</span></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c28">x2i</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></i>;<br/>
<a NAME="E32:35_1_4_1"/><b>then </b>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c27">x1i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c28">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c27">x1i</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c28">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal6.html#T6" title="TOPREAL6:th.6">TOPREAL6:6</a></i>;<br/>
<a NAME="E33:35_1_4_1"/><b>then </b>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c27">x1i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c28">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c26">b1i</font>,<font color="Maroon" title="c27">x1i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c28">x2i</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>;<br/>
<a NAME="E34:35_1_4_1"/><b>then </b><i><font color="Green" title="E43">A35</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c27">x1i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c28">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c26">b1i</font>,<font color="Maroon" title="c27">x1i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c29">b2i</font>,<font color="Maroon" title="c28">x2i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>;<br/>
<a NAME="E35:35_1_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c26">b1i</font>,<font color="Maroon" title="c27">x1i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c29">b2i</font>,<font color="Maroon" title="c28">x2i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">r1</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E28:35_1_4_1"><i><font color="Green" title="E40">A32</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E36:35_1_4_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c23">fb</font>,<font color="Maroon" title="c10">fx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">r</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E30:35_1_4_1"><i><font color="Green" title="E42">A34</font></i></a>, <a class="txt" href="topreal7.html#E34:35_1_4_1"><i><font color="Green" title="E43">A35</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E37:35_1_4_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c10">fx</font>,<font color="Maroon" title="c11">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E38:35_1_4_1"/>
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font>
 <b>by </b><i><a class="txt" href="topreal7.html#E7:35_1_4_1"><i><font color="Green" title="E32">A24</font></i></a>, <a class="txt" href="topreal7.html#E21:35_1_4_1"><i><font color="Green" title="E39">A31</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:35_1_4"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal7.html#E4:35_1_4"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:35_1_4"/>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:35_1"/>
<font color="Maroon" title="c7">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topreal7.html#E11:35_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>

<a NAME="E14:35_1"/><i><font color="Green" title="E28">A36</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E15:35_1"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>  st <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">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" title="c8">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:35_1_5"/>
<font color="Maroon" title="c8">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:35_1_5"/><b>then </b>
<font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E3:35_1_5"/><b>then </b><i><font color="Green" title="E29">A37</font></i>: 
<font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:35_1_5"/><i><font color="Green" title="E30">A38</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">P</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E5:35_1_5"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">P</font> holds <br/> ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">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" title="c9">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:35_1_5_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1_5_1"/><i><font color="Green" title="E31">A39</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:35_1_5_1"/><i><font color="Green" title="E32">A40</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font>
 <b>and </b><br/><a NAME="E5:35_1_5_1"/><i><font color="Green" title="E33">A41</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c10">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="topreal7.html#E3:35_1_5_1"><i><font color="Green" title="E31">A39</font></i></a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E8:35_1_5_1"/><i><font color="Green" title="E34">A42</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E9:35_1_5_1"/><i><font color="Green" title="E35">A43</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P</font>
 <b>by </b><i><a class="txt" href="topreal7.html#E3:35_1_5"><i><font color="Green" title="E29">A37</font></i></a>, <a class="txt" href="topreal7.html#E4:35_1_5_1"><i><font color="Green" title="E32">A40</font></i></a>, <a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">r</font>
;<br/>

<b>thus </b><a NAME="E10:35_1_5_1"/>
<font color="Maroon" title="c12">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="topreal7.html#E8:35_1_5_1"><i><font color="Green" title="E34">A42</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c13">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E11:35_1_5_1"/><i><font color="Green" title="E36">A44</font></i>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c12">r</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c14">bb</font> = <font color="Maroon" title="c13">b</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">bb2</font> = <font color="Maroon" title="c14">bb</font>, <font color="Maroon" title="c16">xx2</font> = <font color="Maroon" title="c9">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span> ;<br/>
<a NAME="E14:35_1_5_1"/><i><font color="Green" title="E37">A45</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c14">bb</font>,<font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E11:35_1_5_1"><i><font color="Green" title="E36">A44</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">k</font> = <font color="Maroon" title="c14">bb</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span> ;<br/>
<a NAME="E16:35_1_5_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">g</font>, <font color="Maroon" title="c19">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E18:35_1_5_1"/><i><font color="Green" title="E38">A46</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font>
 <b>and </b><br/><a NAME="E19:35_1_5_1"/><i><font color="Green" title="E39">A47</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font>
 <b>and </b><br/><a NAME="E20:35_1_5_1"/><i><font color="Green" title="E40">A48</font></i>: 
<font color="Maroon" title="c17">k</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c18">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c19">h</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T26" title="FINSEQ_2:th.26">FINSEQ_2:26</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">p</font>, <font color="Maroon" title="c21">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E22:35_1_5_1"/><i><font color="Green" title="E41">A49</font></i>: 
<font color="Maroon" title="c20">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <b>and </b><br/><a NAME="E23:35_1_5_1"/><i><font color="Green" title="E42">A50</font></i>: 
<font color="Maroon" title="c21">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span>
 <b>and </b><br/><a NAME="E24:35_1_5_1"/><i><font color="Green" title="E43">A51</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c21">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="topreal7.html#E4:35_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">p</font> = <font color="Maroon" title="c20">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span> <b>by </b><i><a class="txt" href="topreal7.html#E22:35_1_5_1"><i><font color="Green" title="E41">A49</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">q</font> = <font color="Maroon" title="c21">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span> <b>by </b><i><a class="txt" href="topreal7.html#E23:35_1_5_1"><i><font color="Green" title="E42">A50</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">fi</font>, <font color="Maroon" title="c25">fj</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E28:35_1_5_1"/><i><font color="Green" title="E44">A52</font></i>: 
( <font color="Maroon" title="c24">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p</font> &amp; <font color="Maroon" title="c25">fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q</font> )
 <b>and </b><br/><a NAME="E29:35_1_5_1"/><i><font color="Green" title="E45">A53</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c23">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">fi</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">fj</font>
 <b>by </b><i><a class="txt" href="topreal7.html#E3:35_1"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">gg</font> = <font color="Maroon" title="c18">g</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">i</font></span>)</span> <b>by </b><i><a class="txt" href="topreal7.html#E18:35_1_5_1"><i><font color="Green" title="E38">A46</font></i></a>, <a class="ref" href="topreal7.html#T17" target="_self" title="TOPREAL7:th.17">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">hh</font> = <font color="Maroon" title="c19">h</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">j</font></span>)</span> <b>by </b><i><a class="txt" href="topreal7.html#E19:35_1_5_1"><i><font color="Green" title="E39">A47</font></i></a>, <a class="ref" href="topreal7.html#T17" target="_self" title="TOPREAL7:th.17">Th17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">g</font>, <font color="Maroon" title="c29">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E33:35_1_5_1"/><i><font color="Green" title="E46">A54</font></i>: 
( <font color="Maroon" title="c28">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">gg</font> &amp; <font color="Maroon" title="c29">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">hh</font> )
 <b>and </b><br/><a NAME="E34:35_1_5_1"/><i><font color="Green" title="E47">A55</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c26">gg</font>,<font color="Maroon" title="c27">hh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h</font>
 <b>by </b><i><a class="txt" href="topreal7.html#E3:35_1"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c30">gg2</font> = <font color="Maroon" title="c26">gg</font>, <font color="Maroon" title="c31">a12</font> = <font color="Maroon" title="c11">a</font> <a href="topreal7.html#K3" title="TOPREAL7:func.3">`1</a>  as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">i</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c32">hh2</font> = <font color="Maroon" title="c27">hh</font>, <font color="Maroon" title="c33">a22</font> = <font color="Maroon" title="c11">a</font> <a href="topreal7.html#K4" title="TOPREAL7:func.4">`2</a>  as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">j</font> ;<br/>
<a NAME="E37:35_1_5_1"/><i><font color="Green" title="E48">A56</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E28:35_1_5_1"><i><font color="Green" title="E44">A52</font></i></a>, <a class="txt" href="topreal7.html#E33:35_1_5_1"><i><font color="Green" title="E46">A54</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E38:35_1_5_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">bb2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c16">xx2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E14:35_1_5_1"><i><font color="Green" title="E37">A45</font></i></a>, <a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>;<br/>
<a NAME="E39:35_1_5_1"/><b>then </b>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">bb2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c16">xx2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></i>;<br/>
<a NAME="E40:35_1_5_1"/><b>then </b>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c28">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c24">fi</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c29">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c25">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E5:35_1_5_1"><i><font color="Green" title="E33">A41</font></i></a>, <a class="txt" href="topreal7.html#E20:35_1_5_1"><i><font color="Green" title="E40">A48</font></i></a>, <a class="txt" href="topreal7.html#E24:35_1_5_1"><i><font color="Green" title="E43">A51</font></i></a>, <a class="txt" href="topreal7.html#E29:35_1_5_1"><i><font color="Green" title="E45">A53</font></i></a>, <a class="txt" href="topreal7.html#E33:35_1_5_1"><i><font color="Green" title="E46">A54</font></i></a>, <a class="txt" href="topreal7.html#E37:35_1_5_1"><i><font color="Green" title="E48">A56</font></i></a>, <a class="ref" href="topreal7.html#T16" target="_self" title="TOPREAL7:th.16">Th16</a></i>;<br/>
<a NAME="E41:35_1_5_1"/><b>then </b>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c28">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c24">fi</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c29">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c25">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<b>by </b><i><a class="ref" href="topreal7.html#T11" target="_self" title="TOPREAL7:th.11">Th11</a></i>;<br/>
<a NAME="E42:35_1_5_1"/><b>then </b><i><font color="Green" title="E49">A57</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c28">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c24">fi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c29">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c25">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>;<br/>
<a NAME="E43:35_1_5_1"/><i><font color="Green" title="E50">A58</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c26">gg</font>,<font color="Maroon" title="c27">hh</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span>,<span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="topreal7.html#K3" title="TOPREAL7:func.3">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="topreal7.html#K4" title="TOPREAL7:func.4">`2</a> </span>)</span></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c26">gg</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="topreal7.html#K3" title="TOPREAL7:func.3">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c27">hh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="topreal7.html#K4" title="TOPREAL7:func.4">`2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal7.html#T20" target="_self" title="TOPREAL7:th.20">Th20</a></i>;<br/>
<a NAME="E44:35_1_5_1"/><i><font color="Green" title="E51">A59</font></i>: 
( <font color="Maroon" title="c31">a12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p</font> &amp; <font color="Maroon" title="c33">a22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q</font> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E24:35_1_5_1"><i><font color="Green" title="E43">A51</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E45:35_1_5_1"/><i><font color="Green" title="E52">A60</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c30">gg2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c31">a12</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c32">hh2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c33">a22</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></i>;<br/>
<a NAME="E46:35_1_5_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p2">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c24">fi</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p2">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c25">fj</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></i>;<br/>
<a NAME="E47:35_1_5_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c30">gg2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c31">a12</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c24">fi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c25">fj</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c32">hh2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c33">a22</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c24">fi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c25">fj</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E28:35_1_5_1"><i><font color="Green" title="E44">A52</font></i></a>, <a class="txt" href="topreal7.html#E33:35_1_5_1"><i><font color="Green" title="E46">A54</font></i></a>, <a class="txt" href="topreal7.html#E44:35_1_5_1"><i><font color="Green" title="E51">A59</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E48:35_1_5_1"/><b>then </b>
(  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c30">gg2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c31">a12</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c28">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c24">fi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c29">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c25">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c32">hh2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c33">a22</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c28">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c24">fi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" title="RVSUM_1:func.12">sqr</a> <span class="p4">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p5">(<span class="default"><font color="Maroon" title="c29">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c25">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E45:35_1_5_1"><i><font color="Green" title="E52">A60</font></i></a>, <a class="ref" href="square_1.html#T94" title="SQUARE_1:th.94">SQUARE_1:94</a></i>;<br/>
<a NAME="E49:35_1_5_1"/><b>then </b>
(  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c30">gg2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c31">a12</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font> &amp;  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="euclid.html#K10" title="EUCLID:func.10">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c32">hh2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c33">a22</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E42:35_1_5_1"><i><font color="Green" title="E49">A57</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E50:35_1_5_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">gg2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c31">a12</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">hh2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c33">a22</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></i>;<br/>
<a NAME="E51:35_1_5_1"/><b>then </b><i><font color="Green" title="E53">A61</font></i>: 
( <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c30">gg2</font>,<font color="Maroon" title="c31">a12</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font> &amp; <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c32">hh2</font>,<font color="Maroon" title="c33">a22</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>;<br/>
<a NAME="E52:35_1_5_1"/>
(  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c26">gg</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="topreal7.html#K3" title="TOPREAL7:func.3">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c27">hh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="topreal7.html#K4" title="TOPREAL7:func.4">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c26">gg</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="topreal7.html#K3" title="TOPREAL7:func.3">`1</a> </span>)</span> or  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c26">gg</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="topreal7.html#K3" title="TOPREAL7:func.3">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c27">hh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="topreal7.html#K4" title="TOPREAL7:func.4">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c27">hh</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="topreal7.html#K4" title="TOPREAL7:func.4">`2</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>;<br/>
<a NAME="E53:35_1_5_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c26">gg</font>,<font color="Maroon" title="c27">hh</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span>,<font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E4:35_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="topreal7.html#E43:35_1_5_1"><i><font color="Green" title="E50">A58</font></i></a>, <a class="txt" href="topreal7.html#E51:35_1_5_1"><i><font color="Green" title="E53">A61</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E54:35_1_5_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">g</font>,<font color="Maroon" title="c29">h</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">r</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E33:35_1_5_1"><i><font color="Green" title="E46">A54</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E55:35_1_5_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">g</font>,<font color="Maroon" title="c29">h</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">P</font>
 
<b>by </b><i><a class="txt" href="topreal7.html#E9:35_1_5_1"><i><font color="Green" title="E35">A43</font></i></a></i>;<br/>
<b>hence </b><a NAME="E56:35_1_5_1"/>
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">P</font>
 <b>by </b><i><a class="txt" href="topreal7.html#E20:35_1_5_1"><i><font color="Green" title="E40">A48</font></i></a>, <a class="txt" href="topreal7.html#E33:35_1_5_1"><i><font color="Green" title="E46">A54</font></i></a>, <a class="txt" href="topreal7.html#E34:35_1_5_1"><i><font color="Green" title="E47">A55</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:35_1_5"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal7.html#E4:35_1_5"><i><font color="Green" title="E30">A38</font></i></a>, <a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></i>;<br/>
<a NAME="E7:35_1_5"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<b>hence </b><a NAME="E8:35_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topreal7.html#E9:35_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="topreal7.html#E10:35_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="tops_2.html#T67" title="TOPS_2:th.67">TOPS_2:67</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:35_1"/>
<font color="Maroon" title="c7">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topreal7.html#E14:35_1"><i><font color="Green" title="E28">A36</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:35"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">j</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>, <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="t_0topsp.html#R1" title="T_0TOPSP:pred.1">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="topreal7.html#T25" target="_self" title="TOPREAL7:th.25">Th25</a></i>;<br/>


</div>
