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

<span class="kw">reconsider </span><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">g</font> =  <a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>);<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:101_1_1_1"/>
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">n</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">z</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c4">n</font></span>)</span> ;<br/><a NAME="E3:101_1_1_1_1"/>
 <a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">T</font> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="toprealc.html#T48" target="_self" title="TOPREALC:th.48">Th48</a></span>;<br/><span class="kw">hence </span><a NAME="E4:101_1_1_1_1"/>
 <a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</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="#E1:101_1_1_1_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_1_2"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c4">n</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/><div class="add"><a NAME="E2:101_1_1_1_2"/>
 for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font><br/>  for <font color="Olive" title="b2">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>  st <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>  st <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font> &amp; <font color="Maroon" title="c7">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:101_1_1_1_2_1"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">and </span><br/><a NAME="E2:101_1_1_1_2_1"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c7">V</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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font> )</span><br/>

<a NAME="E3:101_1_1_1_2_1"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E2:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="tops_1.html#T23" title="TOPS_1:th.23">TOPS_1:23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">s</font> = <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</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="c4">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T67" title="EUCLID:th.67">EUCLID:67</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E6:101_1_1_1_2_1"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<font color="Maroon" title="c9">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="E7:101_1_1_1_2_1"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c9">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="goboard6.html#T5" title="GOBOARD6:th.5">GOBOARD6:5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">G</font> = <font color="Maroon" title="c4">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span> 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="E9:101_1_1_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_1_2"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<a NAME="E10:101_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font>
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">F</font> = <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T24" title="EUCLID:th.24">EUCLID:24</a></span>;<br/>
<a NAME="E12:101_1_1_1_2_1"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
<font color="Maroon" title="c11">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealc.html#D4" target="_self" title="TOPREALC:def.4">Def4</a></span>;<br/>
<a NAME="E13:101_1_1_1_2_1"/><span class="lab"><font color="Green" title="E50">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<span class="lab"><font color="Green" title="E51">A11</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">n</font> = 
<font color="Maroon" title="c10">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealc.html#D4" target="_self" title="TOPREALC:def.4">Def4</a></span>
<br/>.= 
<font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E13:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">R</font> = <font color="Maroon" title="c9">r</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">RR</font> =  <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<a NAME="E15:101_1_1_1_2_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_1_2"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E14:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E6:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="topreal6.html#T15" title="TOPREAL6:th.15">TOPREAL6:15</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E17:101_1_1_1_2_1"/><span class="lab"><font color="Green" title="E52">A12</font></span>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W</font> &amp; <font color="Maroon" title="c14">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">and </span><br/><a NAME="E18:101_1_1_1_2_1"/><span class="lab"><font color="Green" title="E53">A13</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c14">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T10" title="JGRAPH_2:th.10">JGRAPH_2:10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">W</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W</font> &amp; <font color="Maroon" title="c14">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c14">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font> )</span><br/>

<span class="kw">thus </span><a NAME="E19:101_1_1_1_2_1"/>
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W</font> &amp; <font color="Maroon" title="c14">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c14">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c15">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice7.html#D1" title="LATTICE7:def.1">LATTICE7:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c14">W</font> or <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E20:101_1_1_1_2_1"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c14">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E22:101_1_1_1_2_1"/><span class="lab"><font color="Green" title="E54">A14</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W</font>
 <span class="kw">and </span><br/><a NAME="E23:101_1_1_1_2_1"/><span class="lab"><font color="Green" title="E55">A15</font></span>: 
<span class="p1">(<span class="default"><a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T65" title="FUNCT_2:th.65">FUNCT_2:65</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">y1</font> = <font color="Maroon" title="c15">y</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="c4">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T67" title="EUCLID:th.67">EUCLID:67</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">y2</font> = <font color="Maroon" title="c17">y1</font>, <font color="Maroon" title="c19">s2</font> = <font color="Maroon" title="c8">s</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="c4">n</font> ;<br/>
<a NAME="E26:101_1_1_1_2_1"/><span class="lab"><font color="Green" title="E56">A16</font></span>: 
<font color="Maroon" title="c18">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="toprealc.html#D4" target="_self" title="TOPREALC:def.4">Def4</a></span>;<br/>
<span class="lab"><font color="Green" title="E57">A17</font></span>: <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">y1</font>,<font color="Maroon" title="c8">s</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="c18">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c19">s2</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/>.= 
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="txt" href="#E12:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="toprealc.html#T11" target="_self" title="TOPREALC:th.11">Th11</a></span>
;<br/>
<a NAME="E28:101_1_1_1_2_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c14">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></span>;<br/>
<a NAME="E29:101_1_1_1_2_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E14:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="ref" href="rcomp_1.html#T1" title="RCOMP_1:th.1">RCOMP_1:1</a></span>;<br/>
<a NAME="E30:101_1_1_1_2_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c17">y1</font>,<font color="Maroon" title="c8">s</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_1_2"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E27:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="ref" href="xreal_1.html#T79" title="XREAL_1:th.79">XREAL_1:79</a></span>;<br/>
<a NAME="E31:101_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">y</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="c8">s</font>,<font color="Maroon" title="c9">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:101_1_1_1_2_1"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_1_1_2_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:101_1_1_1_2"/>
 <a href="toprealc.html#K15" title="TOPREALC:func.15">incl</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</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="ref" href="jgraph_2.html#T10" title="JGRAPH_2:th.10">JGRAPH_2:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
