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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V5">being_T4</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>;<br/>



<b>assume </b><a NAME="E2:29"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<a NAME="E3:29"/><i><font color="Green">E24</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E4:29"/><i><font color="Green">E25</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>suppose </b></a><a NAME="E1:29_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:29_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span>;<br/><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span>

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> 0
<b>by </b><i><a class="ref" href="toprealb.html#D2">TOPREALB:def 2</a></i>
;<br/><a NAME="E5:29_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/><a NAME="E6:29_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:29_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="tietze.html#E5:29_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E3:29"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T8">TOPMETR:8</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/><a NAME="E9:29_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><b>hence </b><a NAME="E10:29_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tietze.html#E3:29_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2"><b>suppose </b></a><a NAME="E1:29_1_2"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> ;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  : <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  </span>}</span> ;<br/><a NAME="E3:29_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><a NAME="E4:29_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  : <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  </span>}</span> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  : <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topreala.html#T28">TOPREALA:28</a></i>;<br/><a NAME="E7:29_1_2"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="tietze.html#E2:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  ;<br/><a NAME="E9:29_1_2"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tietze.html#D1" target="_self">TIETZE:def 1</a><br/>

<b>assume </b><a NAME="E1:29_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:29_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:29_1_2_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:29_1_2_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E5:29_1_2_1"/><b>then </b>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="tietze.html#E3:29_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="tietze.html#E3:29"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E11:29_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>10</sub></font>, <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> 1 <a href="real_1.html#K6">/</a> 3
 <b>and </b><br/><a NAME="E12:29_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 3
 <b>by </b><i><a class="txt" href="tietze.html#E1:29"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tietze.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E13:29_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><a NAME="E14:29_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> implies ex <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K6">+</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) ) );<br/><a NAME="E16:29_1_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>



<a NAME="E1:29_1_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E3:29_1_2_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="tietze.html#E3:29_1_2_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_2_2_1"/>
(  not <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> or <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_2_2_1_1"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/><b>thus </b><a NAME="E2:29_1_2_2_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K6">+</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="tietze.html#E1:29_1_2_2_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="tietze.html#E3:29_1_2_2"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_2_2_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><a NAME="E8:29_1_2_2_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T88">RELAT_1:88</a></i>;<br/><a NAME="E9:29_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="tietze.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E10:29_1_2_2_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E1:29_1_2_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tietze.html#T17" target="_self">Th17</a></i>;<br/><b>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>21</sub></font>;<br/><i><font color="Green">E32</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/><a NAME="E12:29_1_2_2_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K7">-</a> <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="tietze.html#E11:29_1_2_2_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>25</sub></font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/><a NAME="E16:29_1_2_2_1_2"/>
<font color="Maroon">c<sub>26</sub></font> <a href="partfun3.html#K13">-</a> <font color="Maroon">c<sub>27</sub></font> is <a href="pscomp_1.html#V9">continuous</a>
 ;<br/><a NAME="E17:29_1_2_2_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>23</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/><b>set </b><font color="Maroon">c<sub>28</sub></font> = <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>;<br/><a NAME="E18:29_1_2_2_1_2"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/><a NAME="E19:29_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>29</sub></font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E21:29_1_2_2_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>29</sub></font>, <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 3
 <b>and </b><br/><a NAME="E22:29_1_2_2_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 3
 <b>by </b><i><a class="txt" href="tietze.html#E17:29_1_2_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="tietze.html#E1:29"><i><font color="Green">E22</font></i></a>, <a class="txt" href="tietze.html#E10:29_1_2_2_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tietze.html#T19" target="_self">Th19</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>29</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>30</sub></font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/><b>set </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="partfun3.html#K12">+</a> <font color="Maroon">c<sub>31</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="partfun3.html#K12">+</a> <font color="Maroon">c<sub>31</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E26:29_1_2_2_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><a NAME="E27:29_1_2_2_1_2"/>
<font color="Maroon">c<sub>33</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/><a NAME="E28:29_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="partfun3.html#K12">+</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="tietze.html#E26:29_1_2_2_1_2"><i><font color="Green">E36</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="partfun3.html#K12">+</a> <font color="Maroon">c<sub>31</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> ;<br/><b>take </b>
<font color="Maroon">c<sub>34</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/><b>thus </b><a NAME="E30:29_1_2_2_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="tietze.html#E3:29_1_2_2"><i><font color="Green">E29</font></i></a></i>;<br/><b>assume </b><a NAME="E31:29_1_2_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 ;<br/><b>take </b>
<font color="Maroon">c<sub>29</sub></font>
;<br/><b>thus </b><a NAME="E32:29_1_2_2_1_2"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><a NAME="E33:29_1_2_2_1_2"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><b>hence </b><a NAME="E34:29_1_2_2_1_2"/>
<font color="Maroon">c<sub>29</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E21:29_1_2_2_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E35:29_1_2_2_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E36:29_1_2_2_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E37:29_1_2_2_1_2"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><i><font color="Green">E41</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E35:29_1_2_2_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="tietze.html#E36:29_1_2_2_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="tietze.html#E37:29_1_2_2_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="tietze.html#E4:29"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T63">RFUNCT_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T63">RFUNCT_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T101">RELAT_1:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T63">RFUNCT_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T63">RFUNCT_1:63</a></i>
;<br/><b>hence </b><a NAME="E40:29_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E22:29_1_2_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="tietze.html#E33:29_1_2_2_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="tietze.html#E38:29_1_2_2_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tietze.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E18:29_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E19:29_1_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="txt" href="tietze.html#E16:29_1_2"><i><font color="Green">E28</font></i></a>);<br/></i><a NAME="E20:29_1_2"/><i><font color="Green">E31</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:29_1_2"/><i><font color="Green">E32</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:29_1_2_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i><a class="txt" href="tietze.html#E19:29_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><b>hence </b><a NAME="E2:29_1_2_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> is   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="tietze.html#E21:29_1_2"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="seqfunc.html#M1">Functional_Sequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="tietze.html#E20:29_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="seqfunc.html#T1">SEQFUNC:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:29_1_2_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i><a class="txt" href="tietze.html#E19:29_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><b>hence </b><a NAME="E2:29_1_2_4"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E25:29_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="seqfunc.html#R1">common_on_dom</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="seqfunc.html#D10">SEQFUNC:def 10</a></i>;<br/><a NAME="E26:29_1_2"/>
( 2 <a href="real_1.html#K6">/</a> 9 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; 2 <a href="real_1.html#K6">/</a> 3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E27:29_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
( 2 <a href="real_1.html#K6">/</a> 9 is <a href="xxreal_0.html#V2">positive</a> &amp; 2 <a href="real_1.html#K6">/</a> 3 is <a href="xxreal_0.html#V2">positive</a> )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 9</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E29:29_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> 0 &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K6">+</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="tietze.html#E19:29_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E30:29_1_2"/><i><font color="Green">E36</font></i>: 
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="power.html#K4">to_power</a> 0 <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> 0 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E32:29_1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="tietze.html#E29:29_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="tietze.html#E30:29_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="tietze.html#E18:29_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="tietze.html#E12:29_1_2"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E33:29_1_2"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/><i><font color="Green">E39</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/><i><font color="Green">E40</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E34:29_1_2"><i><font color="Green">E39</font></i></a></i>
;<br/><i><font color="Green">E41</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E34:29_1_2"><i><font color="Green">E39</font></i></a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_2_8"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a>, <a class="txt" href="tietze.html#E36:29_1_2"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a>, <a class="txt" href="tietze.html#E35:29_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="tietze.html#E1:29_1_2_8"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font>

;<br/><br/></div><b>end;</b></div><a NAME="E38:29_1_2"/><b>then </b>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 9</span>)</span> <a href="real_1.html#K4">*</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="tietze.html#E33:29_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="tietze.html#E29:29_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="tietze.html#E32:29_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a>, <a class="txt" href="tietze.html#E36:29_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E39:29_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i><a class="txt" href="tietze.html#E18:29_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="tietze.html#E12:29_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tietze.html#E30:29_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tietze.html#T24" target="_self">Th24</a>, <a class="txt" href="tietze.html#E32:29_1_2"><i><font color="Green">E37</font></i></a></i>;<br/><div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E2:29_1_2_9"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K6">+</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="tietze.html#E19:29_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><b>assume </b><a NAME="E3:29_1_2_9"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>16</sub></font>]
 ;<br/><a NAME="E4:29_1_2_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E6:29_1_2_9"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="tietze.html#E2:29_1_2_9"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tietze.html#T24" target="_self">Th24</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tietze.html#E21:29_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tietze.html#E21:29_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/><a NAME="E9:29_1_2_9"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="partfun3.html#K12">+</a> <font color="Maroon">c<sub>20</sub></font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E10:29_1_2_9"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i><a class="txt" href="tietze.html#E6:29_1_2_9"><i><font color="Green">E45</font></i></a>, <a class="ref" href="topreal6.html#T83">TOPREAL6:83</a>, <a class="txt" href="tietze.html#E9:29_1_2_9"><i><font color="Green">E46</font></i></a>, <a class="txt" href="tietze.html#E21:29_1_2"><i><font color="Green">E32</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E12:29_1_2_9"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K6">+</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="tietze.html#E19:29_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E13:29_1_2_9"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E6:29_1_2_9"><i><font color="Green">E45</font></i></a>, <a class="ref" href="power.html#T46">POWER:46</a></i>;<br/><i><font color="Green">E50</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><i><font color="Green">E51</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E14:29_1_2_9"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E14:29_1_2_9"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><i><font color="Green">E52</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E14:29_1_2_9"><i><font color="Green">E50</font></i></a>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><i><font color="Green">E53</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E14:29_1_2_9"><i><font color="Green">E50</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E18:29_1_2_9"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E19:29_1_2_9"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_9_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_2_9_5"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E15:29_1_2_9"><i><font color="Green">E51</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E1:29_1_2_9_5"><i><font color="Green">E55</font></i></a>, <a class="txt" href="tietze.html#E17:29_1_2_9"><i><font color="Green">E53</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E1:29_1_2_9_5"><i><font color="Green">E55</font></i></a>, <a class="txt" href="tietze.html#E19:29_1_2_9"><i><font color="Green">E54</font></i></a>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E1:29_1_2_9_5"><i><font color="Green">E55</font></i></a>, <a class="txt" href="tietze.html#E16:29_1_2_9"><i><font color="Green">E52</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E21:29_1_2_9"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E13:29_1_2_9"><i><font color="Green">E49</font></i></a>, <a class="txt" href="tietze.html#E6:29_1_2_9"><i><font color="Green">E45</font></i></a>, <a class="txt" href="tietze.html#E15:29_1_2_9"><i><font color="Green">E51</font></i></a>, <a class="txt" href="tietze.html#E16:29_1_2_9"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E22:29_1_2_9"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tietze.html#T24" target="_self">Th24</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>22</sub></font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E24:29_1_2_9"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="tietze.html#E12:29_1_2_9"><i><font color="Green">E48</font></i></a>, <a class="txt" href="tietze.html#E21:29_1_2_9"><i><font color="Green">E55</font></i></a>, <a class="ref" href="power.html#T46">POWER:46</a></i>;<br/><i><font color="Green">E58</font></i>: <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 9</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><i><font color="Green">E59</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/><i><font color="Green">E60</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E26:29_1_2_9"><i><font color="Green">E59</font></i></a></i>
;<br/><i><font color="Green">E61</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E26:29_1_2_9"><i><font color="Green">E59</font></i></a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_9_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_2_9_10"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a>, <a class="txt" href="tietze.html#E28:29_1_2_9"><i><font color="Green">E61</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a>, <a class="txt" href="tietze.html#E27:29_1_2_9"><i><font color="Green">E60</font></i></a>, <a class="txt" href="tietze.html#E1:29_1_2_9_10"><i><font color="Green">E62</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>

;<br/><br/></div><b>end;</b></div><a NAME="E30:29_1_2_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="tietze.html#E24:29_1_2_9"><i><font color="Green">E57</font></i></a>, <a class="txt" href="tietze.html#E12:29_1_2_9"><i><font color="Green">E48</font></i></a>, <a class="txt" href="tietze.html#E28:29_1_2_9"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E31:29_1_2_9"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="tietze.html#E10:29_1_2_9"><i><font color="Green">E47</font></i></a>, <a class="txt" href="tietze.html#E22:29_1_2_9"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tietze.html#T24" target="_self">Th24</a>, <a class="txt" href="tietze.html#E24:29_1_2_9"><i><font color="Green">E57</font></i></a>, <a class="txt" href="tietze.html#E25:29_1_2_9"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E41:29_1_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="tietze.html#E39:29_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="tietze.html#E40:29_1_2"><i><font color="Green">E43</font></i></a>);<br/></i><div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><a NAME="E1:29_1_2_10"/>
<font color="Maroon">c<sub>16</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:29_1_2_10"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 9</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E41:29_1_2"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E43:29_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#R3">is_unif_conv_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="tietze.html#E25:29_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="tietze.html#E27:29_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tietze.html#T11" target="_self">Th11</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> =  <a href="seqfunc.html#K11">lim</a> <font color="Maroon">c<sub>13</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="tietze.html#E41:29_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tietze.html#T22" target="_self">Th22</a></i>;<br/><div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><a NAME="E1:29_1_2_11"/>
<font color="Maroon">c<sub>17</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:29_1_2_11"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="seq_1.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E41:29_1_2"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E46:29_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="seqfunc.html#R1">common_on_dom</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="tietze.html#E25:29_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="seqfunc.html#T24">SEQFUNC:24</a></i>;<br/><a NAME="E47:29_1_2"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="ref" href="seq_1.html#D1">SEQ_1:def 1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E48:29_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>4</sub></font>, <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="tietze.html#E4:29"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E49:29_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#R2">is_point_conv_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="tietze.html#E43:29_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="seqfunc.html#T23">SEQFUNC:23</a></i>;<br/><b>then </b><i><font color="Green">E50</font></i>: <font color="Maroon">c<sub>16</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="seqfunc.html#K11">lim</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="seqfunc.html#T25">SEQFUNC:25</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E46:29_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="tietze.html#E27:29_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="tietze.html#E45:29_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="tietze.html#E48:29_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tietze.html#T13" target="_self">Th13</a></i>
;<br/><a NAME="E51:29_1_2"/>
<font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tietze.html#D1" target="_self">TIETZE:def 1</a><br/>

<b>assume </b><a NAME="E1:29_1_2_13"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:29_1_2_13"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="tietze.html#E18:29_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:29_1_2_13"/><b>then </b>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> 0</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> 0,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> 1 <a href="real_1.html#K6">/</a> 3 )
 
<b>by </b><i><a class="txt" href="tietze.html#E11:29_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="tietze.html#E18:29_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E5:29_1_2_13"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 3
 
<b>by </b><i><a class="ref" href="tietze.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:29_1_2_13"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 3 )
 
<b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E7:29_1_2_13"/><b>then </b><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 9</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 9</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 9</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 9</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 9</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 9</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="tietze.html#E25:29_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="tietze.html#E42:29_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="tietze.html#E27:29_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tietze.html#T12" target="_self">Th12</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:29_1_2_13"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="tietze.html#E7:29_1_2_13"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:29_1_2_13"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="tietze.html#T23" target="_self">Th23</a></i>;<br/> <a href="toprealb.html#K5">R^1</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E3:29"><i><font color="Green">E24</font></i></a></i>
;<br/><a NAME="E54:29_1_2"/><b>then </b>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E55:29_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T9">TOPMETR:9</a></i>;<br/><b>hence </b><a NAME="E56:29_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tietze.html#E50:29_1_2"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
