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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</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>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <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><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>, <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">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> </span><br/>



<a NAME="E1:36"/>
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">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#K3" title="PCOMPS_1:func.3">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="36_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> 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="E1:36_1"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
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#K3" title="PCOMPS_1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/>
<a NAME="E2:36_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
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="36_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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="b2">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="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">u</font>]</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Olive" title="b1">u</font>]</span><br/>



<a NAME="E1:36_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/>
<span class="kw">then </span><span class="kw">reconsider </span><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/>
<a NAME="E3:36_1_1"/>
<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> is   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <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>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T127" title="FINSEQ_2:th.127">FINSEQ_2:127</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T151" title="FINSEQ_2:th.151">FINSEQ_2:151</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">u</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>]</span><br/>

<span class="kw">thus </span><a NAME="E5:36_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>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:36_1"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E2:36_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>);<br/></span>
<a NAME="E5:36_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal7.html#D1" target="_self" title="TOPREAL7:def.1">Def1</a></span>;<br/>
<a NAME="E6:36_1"/>
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#K3" title="PCOMPS_1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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#K3" title="PCOMPS_1:func.3">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#K3" title="PCOMPS_1:func.3">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">f</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">V71</a>( <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">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#K3" title="PCOMPS_1:func.3">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>)</span><br/>

<span class="kw">thus </span><a NAME="E8:36_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K3" title="PCOMPS_1:func.3">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="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K3" title="PCOMPS_1:func.3">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> &amp; <font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c7">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <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>  )</span><br/>

<span class="kw">thus </span><a NAME="E9:36_1"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K3" title="PCOMPS_1:func.3">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c7">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <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>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:36_1_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K3" title="PCOMPS_1:func.3">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>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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#K3" title="PCOMPS_1:func.3">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> or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:36_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#K3" title="PCOMPS_1:func.3">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/>
<a NAME="E4:36_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E6:36_1_2"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
(  <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> )
 <span class="kw">and </span><br/><a NAME="E7:36_1_2"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T26" title="FINSEQ_2:th.26">FINSEQ_2:26</a></span>;<br/>
<a NAME="E8:36_1_2"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_1_2"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="topreal7.html#T17" target="_self" title="TOPREAL7:th.17">Th17</a></span>;<br/>
<a NAME="E9:36_1_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E10:36_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:36_1_2"/>
 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> <font color="Maroon" title="c10">g</font> &amp; <font color="Olive" title="b2">fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E8:36_1_2"><span class="lab"><font color="Green" title="E26">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:36_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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36_1_2"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E10:36_1_2"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E10:36_1"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <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>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K33</a>(<font color="Maroon" title="c7">f</font>) or  not <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K33</a>(<font color="Maroon" title="c7">f</font>) or  not <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> or <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K33</a>(<font color="Maroon" title="c7">f</font>) or  not <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> or <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:36_1_3"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:36_1_3"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:36_1_3"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:36_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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E8:36_1_3"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E9:36_1_3"/><span class="lab"><font color="Green" title="E29">A15</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E10:36_1_3"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:36_1_3"/><span class="lab"><font color="Green" title="E31">A17</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font></span><br/>

<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E13:36_1_3"/><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c14">fi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x1a</font>
 <span class="kw">and </span><br/><a NAME="E14:36_1_3"/><span class="lab"><font color="Green" title="E33">A19</font></span>: 
<font color="Maroon" title="c15">fj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1b</font>
 <span class="kw">and </span><br/><a NAME="E15:36_1_3"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E3:36_1_3"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E4:36_1_3"><span class="lab"><font color="Green" title="E26">A12</font></span></a></span>;<br/>
<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E17:36_1_3"/><span class="lab"><font color="Green" title="E35">A21</font></span>: 
<font color="Maroon" title="c16">fi2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2a</font>
 <span class="kw">and </span><br/><a NAME="E18:36_1_3"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
<font color="Maroon" title="c17">fj2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2b</font>
 <span class="kw">and </span><br/><a NAME="E19:36_1_3"/><span class="lab"><font color="Green" title="E37">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E8:36_1_3"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E9:36_1_3"><span class="lab"><font color="Green" title="E29">A15</font></span></a></span>;<br/>
<a NAME="E20:36_1_3"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1_3"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E14:36_1_3"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E21:36_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A24</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36_1_3"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E18:36_1_3"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E22:36_1_3"/><span class="lab"><font color="Green" title="E39">A25</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1_3"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E13:36_1_3"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E23:36_1_3"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36_1_3"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E17:36_1_3"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E24:36_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">fi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">fi2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1_3"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E10:36_1_3"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E15:36_1_3"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E19:36_1_3"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E21:36_1_3"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E11:36_1_3"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="topreal7.html#T6" target="_self" title="TOPREAL7:th.6">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:36_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1_3"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E10:36_1_3"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E13:36_1_3"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E14:36_1_3"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E15:36_1_3"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E17:36_1_3"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="#E18:36_1_3"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E19:36_1_3"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E22:36_1_3"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E21:36_1_3"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E11:36_1_3"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="topreal7.html#T6" target="_self" title="TOPREAL7:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E11:36_1"/><span class="lab"><font color="Green" title="E25">A26</font></span>: 
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#K3" title="PCOMPS_1:func.3">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#K8" title="RELSET_1:func.8">"</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="36_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K3" title="PCOMPS_1:func.3">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_1_4"/>
<font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E2:36_1_4"/><span class="kw">then </span>
<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#K3" title="PCOMPS_1:func.3">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="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E3:36_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A27</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/>
<a NAME="E4:36_1_4"/><span class="lab"><font color="Green" title="E27">A28</font></span>: 
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#K7" title="RELSET_1:func.7">.:</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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">P</font> implies  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">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="Maroon" title="c9">x</font>,<font color="Olive" title="b1">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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">P</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_1_4_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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">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="Maroon" title="c9">x</font>,<font color="Olive" title="b1">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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">P</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:36_1_4_1"/><span class="lab"><font color="Green" title="E28">A29</font></span>: 
<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#K3" title="PCOMPS_1:func.3">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="kw">and </span><br/><a NAME="E4:36_1_4_1"/><span class="lab"><font color="Green" title="E29">A30</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:36_1_4_1"/><span class="lab"><font color="Green" title="E30">A31</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1_4_1"><span class="lab"><font color="Green" title="E28">A29</font></span></a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E8:36_1_4_1"/><span class="lab"><font color="Green" title="E31">A32</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E9:36_1_4_1"/><span class="lab"><font color="Green" title="E32">A33</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1_4"><span class="lab"><font color="Green" title="E26">A27</font></span></a>, <a class="txt" href="#E4:36_1_4_1"><span class="lab"><font color="Green" title="E29">A30</font></span></a>, <a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">r</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  &amp;  <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">P</font> )</span><br/>

<span class="kw">thus </span><a NAME="E10:36_1_4_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36_1_4_1"><span class="lab"><font color="Green" title="E31">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">P</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">P</font> )</span><br/>

<span class="kw">assume </span><a NAME="E11:36_1_4_1"/><span class="lab"><font color="Green" title="E33">A34</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">P</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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/>
<span class="kw">reconsider </span><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="E14:36_1_4_1"/>
 <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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1_4_1"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>;<br/>
<a NAME="E15:36_1_4_1"/><span class="kw">then </span>
<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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></span>;<br/>
<a NAME="E16:36_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A35</font></span>: 
 <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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>;<br/>
<span class="kw">reconsider </span><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="E18:36_1_4_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E20:36_1_4_1"/><span class="lab"><font color="Green" title="E35">A36</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E21:36_1_4_1"/><span class="lab"><font color="Green" title="E36">A37</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E22:36_1_4_1"/><span class="lab"><font color="Green" title="E37">A38</font></span>: 
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T26" title="FINSEQ_2:th.26">FINSEQ_2:26</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:36_1_4_1"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="ref" href="topreal7.html#T17" target="_self" title="TOPREAL7:th.17">Th17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_1_4_1"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="ref" href="topreal7.html#T17" target="_self" title="TOPREAL7:th.17">Th17</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">g</font>, <font color="Maroon" title="c23">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> <span class="kw"> such that </span><br/><a NAME="E26:36_1_4_1"/><span class="lab"><font color="Green" title="E38">A39</font></span>: 
<font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">gg</font>
 <span class="kw">and </span><br/><a NAME="E27:36_1_4_1"/><span class="lab"><font color="Green" title="E39">A40</font></span>: 
<font color="Maroon" title="c23">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">hh</font>
 <span class="kw">and </span><br/><a NAME="E28:36_1_4_1"/><span class="lab"><font color="Green" title="E40">A41</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c21">gg</font>,<font color="Maroon" title="c20">hh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c23">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">gg2</font> = <font color="Maroon" title="c21">gg</font>, <font color="Maroon" title="c25">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/>
<a NAME="E30:36_1_4_1"/><span class="lab"><font color="Green" title="E41">A42</font></span>: 
(  <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="c21">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="c20">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="c21">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="c21">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="c20">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="c20">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">p</font>, <font color="Maroon" title="c27">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E32:36_1_4_1"/><span class="lab"><font color="Green" title="E42">A43</font></span>: 
<font color="Maroon" title="c26">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>
 <span class="kw">and </span><br/><a NAME="E33:36_1_4_1"/><span class="lab"><font color="Green" title="E43">A44</font></span>: 
<font color="Maroon" title="c27">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>
 <span class="kw">and </span><br/><a NAME="E34:36_1_4_1"/><span class="lab"><font color="Green" title="E44">A45</font></span>: 
<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="c26">p</font>,<font color="Maroon" title="c27">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">q</font> = <font color="Maroon" title="c27">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:36_1_4_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c29">p</font> = <font color="Maroon" title="c26">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:36_1_4_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c30">fi</font>, <font color="Maroon" title="c31">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> <span class="kw"> such that </span><br/><a NAME="E38:36_1_4_1"/><span class="lab"><font color="Green" title="E45">A46</font></span>: 
<font color="Maroon" title="c30">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p</font>
 <span class="kw">and </span><br/><a NAME="E39:36_1_4_1"/><span class="lab"><font color="Green" title="E46">A47</font></span>: 
<font color="Maroon" title="c31">fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q</font>
 <span class="kw">and </span><br/><a NAME="E40:36_1_4_1"/><span class="lab"><font color="Green" title="E47">A48</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c29">p</font>,<font color="Maroon" title="c28">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">fi</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c31">fj</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/>
<a NAME="E41:36_1_4_1"/><span class="lab"><font color="Green" title="E48">A49</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">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="c31">fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:36_1_4_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E39:36_1_4_1"><span class="lab"><font color="Green" title="E46">A47</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E42:36_1_4_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">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="c23">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:36_1_4_1"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E27:36_1_4_1"><span class="lab"><font color="Green" title="E39">A40</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E43:36_1_4_1"/><span class="kw">then </span>
 <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="c22">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c30">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="c23">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c31">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1_4_1"><span class="lab"><font color="Green" title="E30">A31</font></span></a>, <a class="txt" href="#E22:36_1_4_1"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E34:36_1_4_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E40:36_1_4_1"><span class="lab"><font color="Green" title="E47">A48</font></span></a>, <a class="txt" href="#E26:36_1_4_1"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E27:36_1_4_1"><span class="lab"><font color="Green" title="E39">A40</font></span></a>, <a class="txt" href="#E41:36_1_4_1"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="txt" href="#E16:36_1_4_1"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="ref" href="topreal7.html#T16" target="_self" title="TOPREAL7:th.16">Th16</a></span>;<br/>
<a NAME="E44:36_1_4_1"/><span class="kw">then </span>
 <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="c22">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c30">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="c23">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c31">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal7.html#T11" target="_self" title="TOPREAL7:th.11">Th11</a></span>;<br/>
<a NAME="E45:36_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A50</font></span>: 
 <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="c22">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c30">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="c23">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c31">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c32">hh2</font> = <font color="Maroon" title="c20">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="E47:36_1_4_1"/><span class="lab"><font color="Green" title="E50">A51</font></span>: 
<font color="Maroon" title="c33">a22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:36_1_4_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E48:36_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="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="c22">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c30">fi</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></span>;<br/>
<a NAME="E49:36_1_4_1"/><span class="kw">then </span>
(  <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> &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="c22">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c30">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="c23">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c31">fj</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:36_1_4_1"><span class="lab"><font color="Green" title="E46">A47</font></span></a>, <a class="txt" href="#E27:36_1_4_1"><span class="lab"><font color="Green" title="E39">A40</font></span></a>, <a class="txt" href="#E47:36_1_4_1"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E50:36_1_4_1"/><span class="kw">then </span>
 <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="c22">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c30">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="c23">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c31">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T94" title="SQUARE_1:th.94">SQUARE_1:94</a></span>;<br/>
<a NAME="E51:36_1_4_1"/><span class="kw">then </span>
 <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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:36_1_4_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E52:36_1_4_1"/><span class="kw">then </span>
<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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>;<br/>
<a NAME="E53:36_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A52</font></span>: 
<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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></span>;<br/>
<a NAME="E54:36_1_4_1"/><span class="lab"><font color="Green" title="E52">A53</font></span>: 
<font color="Maroon" title="c25">a12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:36_1_4_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E55:36_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="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="c23">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c31">fj</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></span>;<br/>
<a NAME="E56:36_1_4_1"/><span class="kw">then </span>
(  <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="c24">gg2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">a12</font></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="c24">gg2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">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="c22">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c30">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="c23">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c31">fj</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:36_1_4_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E26:36_1_4_1"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E54:36_1_4_1"><span class="lab"><font color="Green" title="E52">A53</font></span></a>, <a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E57:36_1_4_1"/><span class="kw">then </span>
 <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">gg2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">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="c22">g</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c30">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="c23">h</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c31">fj</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T94" title="SQUARE_1:th.94">SQUARE_1:94</a></span>;<br/>
<a NAME="E58:36_1_4_1"/><span class="kw">then </span>
 <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">gg2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">a12</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:36_1_4_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E59:36_1_4_1"/><span class="kw">then </span>
<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">gg2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">a12</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>;<br/>
<a NAME="E60:36_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A54</font></span>: 
<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="c24">gg2</font>,<font color="Maroon" title="c25">a12</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></span>;<br/>
<a NAME="E61:36_1_4_1"/>
 <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="c21">gg</font>,<font color="Maroon" title="c20">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="c21">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="c20">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal7.html#T20" target="_self" title="TOPREAL7:th.20">Th20</a></span>;<br/>
<a NAME="E62:36_1_4_1"/><span class="kw">then </span>
 <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="c21">gg</font>,<font color="Maroon" title="c20">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E60:36_1_4_1"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="txt" href="#E53:36_1_4_1"><span class="lab"><font color="Green" title="E51">A52</font></span></a>, <a class="txt" href="#E30:36_1_4_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></span>;<br/>
<a NAME="E63:36_1_4_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c23">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:36_1_4_1"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E27:36_1_4_1"><span class="lab"><font color="Green" title="E39">A40</font></span></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>;<br/>
<a NAME="E64:36_1_4_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c23">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36_1_4_1"><span class="lab"><font color="Green" title="E32">A33</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E65:36_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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:36_1_4_1"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E26:36_1_4_1"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E27:36_1_4_1"><span class="lab"><font color="Green" title="E39">A40</font></span></a>, <a class="txt" href="#E28:36_1_4_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:36_1_4"/>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/>
<a NAME="E6:36_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1_4"><span class="lab"><font color="Green" title="E27">A28</font></span></a>, <a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></span>;<br/>
<a NAME="E7:36_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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#K3" title="PCOMPS_1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:36_1_4"/>
<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#K8" title="RELSET_1:func.8">"</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#K3" title="PCOMPS_1:func.3">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E10:36_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="tops_2.html#T67" title="TOPS_2:th.67">TOPS_2:67</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:36_1"/><span class="lab"><font color="Green" title="E26">A55</font></span>: 
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#K3" title="PCOMPS_1:func.3">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#K8" title="RELSET_1:func.8">"</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="36_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K3" title="PCOMPS_1:func.3">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_1_5"/>
<font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E2:36_1_5"/><span class="kw">then </span>
<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#K3" title="PCOMPS_1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E3:36_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A56</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/>
<a NAME="E4:36_1_5"/><span class="lab"><font color="Green" title="E28">A57</font></span>: 
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#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> implies  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">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="Maroon" title="c9">x</font>,<font color="Olive" title="b1">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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_1_5_1"/><span class="lab"><font color="Green" title="E29">A58</font></span>: 
<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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">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="Maroon" title="c9">x</font>,<font color="Olive" title="b1">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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> )</span><br/>

<a NAME="E2:36_1_5_1"/><span class="kw">then </span>
<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#K3" title="PCOMPS_1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/>
<a NAME="E4:36_1_5_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_5_1"><span class="lab"><font color="Green" title="E29">A58</font></span></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:36_1_5_1"/><span class="lab"><font color="Green" title="E30">A59</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E7:36_1_5_1"/><span class="lab"><font color="Green" title="E31">A60</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1_5"><span class="lab"><font color="Green" title="E27">A56</font></span></a>, <a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></span>;<br/>
<span class="kw">take </span><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; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  &amp;  <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> )</span><br/>

<span class="kw">thus </span><a NAME="E8:36_1_5_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_1_5_1"><span class="lab"><font color="Green" title="E30">A59</font></span></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> )</span><br/>

<span class="kw">assume </span><a NAME="E9:36_1_5_1"/><span class="lab"><font color="Green" title="E32">A61</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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/>
<a NAME="E11:36_1_5_1"/><span class="lab"><font color="Green" title="E33">A62</font></span>: 
 <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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36_1_5_1"><span class="lab"><font color="Green" title="E32">A61</font></span></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>;<br/>
<a NAME="E12:36_1_5_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="E13:36_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A63</font></span>: 
<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#K3" title="PCOMPS_1:func.3">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="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/>
<a NAME="E14:36_1_5_1"/><span class="kw">then </span>
<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#K3" title="PCOMPS_1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">fbb</font> = <font color="Maroon" title="c15">fb</font>, <font color="Maroon" title="c17">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/>
<span class="kw">consider </span><font color="Maroon" title="c18">b1</font>, <font color="Maroon" title="c19">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="c20">b2</font>, <font color="Maroon" title="c21">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><span class="kw"> such that </span><br/><a NAME="E18:36_1_5_1"/><span class="lab"><font color="Green" title="E35">A64</font></span>: 
( <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="c18">b1</font>,<font color="Maroon" title="c20">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="c19">x1</font>,<font color="Maroon" title="c21">x2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> )
 <span class="kw">and </span><br/><a NAME="E19:36_1_5_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="c18">b1</font>,<font color="Maroon" title="c19">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="c20">b2</font>,<font color="Maroon" title="c21">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal7.html#D1" target="_self" title="TOPREAL7:def.1">Def1</a></span>;<br/>
<a NAME="E20:36_1_5_1"/><span class="lab"><font color="Green" title="E36">A65</font></span>: 
 <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="c18">b1</font>,<font color="Maroon" title="c20">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="c19">x1</font>,<font color="Maroon" title="c21">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="c18">b1</font>,<font color="Maroon" title="c19">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="c20">b2</font>,<font color="Maroon" title="c21">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal7.html#T20" target="_self" title="TOPREAL7:th.20">Th20</a></span>;<br/>
<a NAME="E21:36_1_5_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">b2</font>,<font color="Maroon" title="c21">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="c18">b1</font>,<font color="Maroon" title="c19">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="c20">b2</font>,<font color="Maroon" title="c21">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E22:36_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A66</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">b2</font>,<font color="Maroon" title="c21">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:36_1_5_1"><span class="lab"><font color="Green" title="E35">A64</font></span></a>, <a class="txt" href="#E20:36_1_5_1"><span class="lab"><font color="Green" title="E36">A65</font></span></a>, <a class="txt" href="#E11:36_1_5_1"><span class="lab"><font color="Green" title="E33">A62</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">x2i</font> = <font color="Maroon" title="c21">x2</font>, <font color="Maroon" title="c23">b2i</font> = <font color="Maroon" title="c20">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">b1i</font> = <font color="Maroon" title="c18">b1</font>, <font color="Maroon" title="c25">x1i</font> = <font color="Maroon" title="c19">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/>
<span class="kw">consider </span><font color="Maroon" title="c26">b1f</font>, <font color="Maroon" title="c27">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> <span class="kw"> such that </span><br/><a NAME="E26:36_1_5_1"/><span class="lab"><font color="Green" title="E38">A67</font></span>: 
( <font color="Maroon" title="c26">b1f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b1</font> &amp; <font color="Maroon" title="c27">b2f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b2</font> )
 <span class="kw">and </span><br/><a NAME="E27:36_1_5_1"/><span class="lab"><font color="Green" title="E39">A68</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c18">b1</font>,<font color="Maroon" title="c20">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">b1f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c27">b2f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/>
<a NAME="E28:36_1_5_1"/><span class="lab"><font color="Green" title="E40">A69</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">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="c27">b2f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:36_1_5_1"><span class="lab"><font color="Green" title="E38">A67</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E29:36_1_5_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c18">b1</font>,<font color="Maroon" title="c19">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="c18">b1</font>,<font color="Maroon" title="c19">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="c20">b2</font>,<font color="Maroon" title="c21">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E30:36_1_5_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c18">b1</font>,<font color="Maroon" title="c19">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:36_1_5_1"><span class="lab"><font color="Green" title="E35">A64</font></span></a>, <a class="txt" href="#E20:36_1_5_1"><span class="lab"><font color="Green" title="E36">A65</font></span></a>, <a class="txt" href="#E11:36_1_5_1"><span class="lab"><font color="Green" title="E33">A62</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E31:36_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A70</font></span>: 
<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="c24">b1i</font>,<font color="Maroon" title="c25">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="c23">b2i</font>,<font color="Maroon" title="c22">x2i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:36_1_5_1"><span class="lab"><font color="Green" title="E37">A66</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E32:36_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="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">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="c23">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c22">x2i</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></span>;<br/>
<a NAME="E33:36_1_5_1"/><span class="kw">then </span>
 <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="c24">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">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="c23">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c22">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="c24">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">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="c23">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c22">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T6" title="TOPREAL6:th.6">TOPREAL6:6</a></span>;<br/>
<a NAME="E34:36_1_5_1"/><span class="kw">then </span>
 <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="c24">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">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="c23">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c22">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="c24">b1i</font>,<font color="Maroon" title="c25">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="c23">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c22">x2i</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></span>;<br/>
<a NAME="E35:36_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A71</font></span>: 
 <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="c24">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">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="c23">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c22">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="c24">b1i</font>,<font color="Maroon" title="c25">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="c23">b2i</font>,<font color="Maroon" title="c22">x2i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">x1f</font>, <font color="Maroon" title="c29">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> <span class="kw"> such that </span><br/><a NAME="E37:36_1_5_1"/><span class="lab"><font color="Green" title="E43">A72</font></span>: 
( <font color="Maroon" title="c28">x1f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x1</font> &amp; <font color="Maroon" title="c29">x2f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x2</font> )
 <span class="kw">and </span><br/><a NAME="E38:36_1_5_1"/><span class="lab"><font color="Green" title="E44">A73</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">x1</font>,<font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">x1f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">x2f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/>
<a NAME="E39:36_1_5_1"/><span class="lab"><font color="Green" title="E45">A74</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">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="c29">x2f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:36_1_5_1"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<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="c16">fbb</font>,<font color="Maroon" title="c17">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="c16">fbb</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c17">fxx</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></span>
<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="c16">fbb</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c17">fxx</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>
<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="c24">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">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="c23">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c22">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:36_1_5_1"><span class="lab"><font color="Green" title="E35">A64</font></span></a>, <a class="txt" href="#E26:36_1_5_1"><span class="lab"><font color="Green" title="E38">A67</font></span></a>, <a class="txt" href="#E27:36_1_5_1"><span class="lab"><font color="Green" title="E39">A68</font></span></a>, <a class="txt" href="#E37:36_1_5_1"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="txt" href="#E38:36_1_5_1"><span class="lab"><font color="Green" title="E44">A73</font></span></a>, <a class="txt" href="#E28:36_1_5_1"><span class="lab"><font color="Green" title="E40">A69</font></span></a>, <a class="txt" href="#E39:36_1_5_1"><span class="lab"><font color="Green" title="E45">A74</font></span></a>, <a class="ref" href="topreal7.html#T16" target="_self" title="TOPREAL7:th.16">Th16</a></span>
<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="c24">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">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="c23">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c22">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal7.html#T11" target="_self" title="TOPREAL7:th.11">Th11</a></span>
<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="c24">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">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="c23">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c22">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></span>
<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="c24">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">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="c23">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c22">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>
<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="c24">b1i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c25">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="c23">b2i</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c22">x2i</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>
;<br/>
<a NAME="E41:36_1_5_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c15">fb</font>,<font color="Maroon" title="c10">fx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:36_1_5_1"><span class="lab"><font color="Green" title="E42">A71</font></span></a>, <a class="txt" href="#E31:36_1_5_1"><span class="lab"><font color="Green" title="E41">A70</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E42:36_1_5_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E43:36_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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36_1_5_1"><span class="lab"><font color="Green" title="E31">A60</font></span></a>, <a class="txt" href="#E13:36_1_5_1"><span class="lab"><font color="Green" title="E34">A63</font></span></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:36_1_5"/>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>;<br/>
<a NAME="E6:36_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1_5"><span class="lab"><font color="Green" title="E28">A57</font></span></a>, <a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:36_1_5"/>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K3" title="PCOMPS_1:func.3">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="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:36_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">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/>
<span class="kw">hence </span><a NAME="E14:36_1"/>
<font color="Maroon" title="c7">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36_1"><span class="lab"><font color="Green" title="E26">A55</font></span></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> </span><br/>

<a NAME="E15:36_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">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/>
<span class="kw">hence </span><a NAME="E16:36_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E25">A26</font></span></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:36"/>
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</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>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <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><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>, <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal7.html#T25" target="_self" title="TOPREAL7:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
