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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c7">Q</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c8">PP</font>, <font color="Maroon" title="c9">QQ</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>







<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c8">PP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c9">QQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c5">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c8">PP</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Q</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c10">IT</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font></span>)</span></span>)</span>;<br/>
<a NAME="E2:25"/><i><font color="Green" title="E22">A2</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></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="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <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 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  )
 
;<br/>
<a NAME="E3:25"/><b>then </b><i><font color="Green" title="E23">A3</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E4:25"/>
<font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
;<br/>
<a NAME="E5:25"/><b>then </b><i><font color="Green" title="E24">A4</font></i>: 
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>;<br/>
<a NAME="E6:25"/><i><font color="Green" title="E25">A5</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
;<br/>
<a NAME="E7:25"/>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="weierstr.html#T17" title="WEIERSTR:th.17">WEIERSTR:17</a></i>;<br/>
<a NAME="E8:25"/><b>then </b><i><font color="Green" title="E26">A6</font></i>: 
(  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> &amp;  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> )
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a>, <a class="ref" href="weierstr.html#T18" title="WEIERSTR:th.18">WEIERSTR:18</a>, <a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>;<br/>
<a NAME="E9:25"/><b>then </b><i><font color="Green" title="E27">A7</font></i>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T30" title="RCOMP_1:th.30">RCOMP_1:30</a></i>;<br/>
<a NAME="E10:25"/><b>then </b><i><font color="Green" title="E28">A8</font></i>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font></span>)</span> <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="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E5:25"><i><font color="Green" title="E24">A4</font></i></a></i>;<br/>
<a NAME="E11:25"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font></span>)</span></span>)</span> <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="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E5:25"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="jordan5a.html#E9:25"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">IT</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font></span>)</span></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan5a.html#E3:25"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E13:25"/><i><font color="Green" title="E29">A9</font></i>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E10:25"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:25"/><i><font color="Green" title="E30">A10</font></i>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="jordan5a.html#E9:25"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>;<br/>
<a NAME="E15:25"/><b>then </b><i><font color="Green" title="E31">A11</font></i>: 
<font color="Maroon" title="c11">IT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">QQ</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="jordan5a.html#E13:25"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E16:25"/>
<font color="Maroon" title="c6">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="jordan5a.html#E6:25"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="compts_1.html#T11" title="COMPTS_1:th.11">COMPTS_1:11</a></i>;<br/>
<a NAME="E17:25"/><b>then </b>
for <font color="Olive" title="b1">P1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>  st <font color="Olive" title="b1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">QQ</font> holds <br/><font color="Olive" title="b1">P1</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="weierstr.html#T14" title="WEIERSTR:th.14">WEIERSTR:14</a></i>;<br/>
<a NAME="E18:25"/><b>then </b>
<font color="Maroon" title="c9">QQ</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="jordan5a.html#E2:25"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="compts_1.html#T11" title="COMPTS_1:th.11">COMPTS_1:11</a></i>;<br/>
<a NAME="E19:25"/><b>then </b>
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c9">QQ</font> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 
<b>by </b><i><a class="ref" href="weierstr.html#T17" title="WEIERSTR:th.17">WEIERSTR:17</a></i>;<br/>
<a NAME="E20:25"/><b>then </b><i><font color="Green" title="E32">A12</font></i>: 
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c9">QQ</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E21:25"/><i><font color="Green" title="E33">A13</font></i>: 
<font color="Maroon" title="c11">IT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c9">QQ</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E15:25"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>;<br/>
<a NAME="E22:25"/><i><font color="Green" title="E34">A14</font></i>: 
for <font color="Olive" title="b1">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>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c9">QQ</font> holds <br/><font color="Maroon" title="c11">IT</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c9">QQ</font>
 ;<br/>

<a NAME="E2:25_1"/><b>then </b>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">P</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>;<br/>
<a NAME="E3:25_1"/><b>then </b>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:25_1"/><i><font color="Green" title="E35">A15</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font> &amp; <font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c15">x2</font> =  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font></span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <b>by </b><i><a class="txt" href="jordan5a.html#E14:25"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="jordan5a.html#E5:25_1"><i><font color="Green" title="E35">A15</font></i></a></i>;<br/>
<a NAME="E7:25_1"/>
( <font color="Maroon" title="c14">x1</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="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> &amp; <font color="Maroon" title="c15">x2</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="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c17">r2</font> = <font color="Maroon" title="c15">x2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan5a.html#E3:25"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E9:25_1"/>
( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</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="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x2</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="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">fr</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x2</font>, <font color="Maroon" title="c19">fx</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan5a.html#E3:25"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E11:25_1"/><i><font color="Green" title="E36">A16</font></i>: 
<font color="Maroon" title="c17">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c16">r1</font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E8:25"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="jordan5a.html#E5:25_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1"/>
( <font color="Maroon" title="c17">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">r1</font> or <font color="Maroon" title="c17">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1"/>
<font color="Maroon" title="c17">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">r1</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">r1</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E11:25_1"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:25_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">fr</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c19">fx</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="jordan5a.html#T16" target="_self" title="JORDAN5A:th.16">Th16</a></i>;<br/><b>hence </b><a NAME="E4:25_1_1_1"/>
<font color="Maroon" title="c11">IT</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">r</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E5:25_1"><i><font color="Green" title="E35">A15</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_2"/>
<font color="Maroon" title="c17">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_2"/>
<font color="Maroon" title="c11">IT</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">r</font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E5:25_1"><i><font color="Green" title="E35">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E23:25"/>
for <font color="Olive" title="b1">s</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>   st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> ex <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c9">QQ</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">IT</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2"/>
 ex <font color="Olive" title="b1">s</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>  st <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">s</font> &amp; ( for <font color="Olive" title="b2">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>   holds <br/>(  not <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c9">QQ</font> or  not <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">IT</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font> ) ) )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">s</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> <b> such that </b><br/><a NAME="E3:25_2"/><i><font color="Green" title="E35">A17</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">s</font> &amp; ( for <font color="Olive" title="b1">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>   holds <br/>(  not <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c9">QQ</font> or  not <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">IT</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> ) ) )
 ;<br/>
<a NAME="E4:25_2"/>
<font color="Maroon" title="c11">IT</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">IT</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0
 
<b>by </b><i><a class="txt" href="jordan5a.html#E3:25_2"><i><font color="Green" title="E35">A17</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<b>hence </b><a NAME="E5:25_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan5a.html#E21:25"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="jordan5a.html#E3:25_2"><i><font color="Green" title="E35">A17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E24:25"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c8">PP</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <font color="Maroon" title="c9">QQ</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan5a.html#E20:25"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="jordan5a.html#E21:25"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="jordan5a.html#E22:25"><i><font color="Green" title="E34">A14</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>


</div>
