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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V5" title="COMPTS_1:attr.5">being_T4</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">A</font> be  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>;<br/>



<b>assume </b><a NAME="E2:29"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>

<a NAME="E3:29"/><i><font color="Green" title="E24">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E4:29"/><i><font color="Green" title="E25">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">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" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">A1</font> = <font color="Maroon" title="c2">A</font> as  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E3:29_1_1"/><i><font color="Green" title="E26">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>set </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/><a NAME="E4:29_1_1"/>
<font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="toprealb.html#D2" title="TOPREALB:def.2">TOPREALB:def 2</a></i>;<br/><a NAME="E5:29_1_1"/><b>then </b><i><font color="Green" title="E27">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/><a NAME="E6:29_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:29_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="tietze.html#E5:29_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E3:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c6">g</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T8" title="TOPMETR:th.8">TOPMETR:8</a></i>;<br/><b>take </b>
<font color="Maroon" title="c7">g</font>
;<br/><a NAME="E9:29_1_1"/>
<font color="Maroon" title="c7">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">A1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><b>hence </b><a NAME="E10:29_1_1"/>
<font color="Maroon" title="c7">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="tietze.html#E3:29_1_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">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" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">A1</font> = <font color="Maroon" title="c2">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/><b>set </b><font color="Maroon" title="c5">DF</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/><b>set </b><font color="Maroon" title="c6">D</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  : <font color="Olive" title="b1">q</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  </span>}</span> ;<br/><a NAME="E3:29_1_2"/>
<font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/><a NAME="E4:29_1_2"/><b>then </b>
<font color="Maroon" title="c1">T</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  : <font color="Olive" title="b1">q</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  </span>}</span> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">D</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  : <font color="Olive" title="b1">q</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c3">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A1</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topreala.html#T28" title="TOPREALA:th.28">TOPREALA:28</a></i>;<br/><a NAME="E7:29_1_2"/>
<font color="Maroon" title="c8">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="tietze.html#E2:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c3">f</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A1</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/><a NAME="E9:29_1_2"/>
<font color="Maroon" title="c9">f1</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">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" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tietze.html#D1" target="_self" title="TIETZE:def.1">TIETZE:def 1</a><br/>

<b>assume </b><a NAME="E1:29_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f1</font></span>)</span>
 ;<br/>

<a NAME="E2:29_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:29_1_2_1"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c9">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:29_1_2_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E5:29_1_2_1"/><b>then </b>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c9">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="tietze.html#E3:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="tietze.html#E3:29_1_2_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_2_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c10">g0</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E11:29_1_2"/><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c10">g0</font>, <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">g0</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3
 <b>and </b><br/><a NAME="E12:29_1_2"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon" title="c9">f1</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c10">g0</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3
 <b>by </b><i><a class="txt" href="tietze.html#E1:29"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="tietze.html#T19" target="_self" title="TIETZE:th.19">Th19</a></i>;<br/><a NAME="E13:29_1_2"/>
<font color="Maroon" title="c10">g0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/><a NAME="E14:29_1_2"/><b>then </b>
<font color="Maroon" title="c10">g0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">g0d</font> = <font color="Maroon" title="c10">g0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">D</font> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">E2</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <br/>( <font color="Olive" title="b1">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; ( <font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Olive" title="b1">E2</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> $1</span>)</span> implies  ex <font color="Olive" title="b2">g</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <br/>( $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">g</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Olive" title="b1">E2</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Olive" title="b2">g</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) ) );<br/><a NAME="E16:29_1_2"/><i><font color="Green" title="E28">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">D</font>  ex <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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" title="c12">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">D</font>;<br/>



<a NAME="E1:29_1_2_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c14">E2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E3:29_1_2_2"/><i><font color="Green" title="E29">A11</font></i>: 
( <font color="Maroon" title="c14">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c14">E2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">E2</font> = <font color="Maroon" title="c14">E2</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="tietze.html#E3:29_1_2_2"><i><font color="Green" title="E29">A11</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" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c15">E2</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</font></span>)</span> or <font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c15">E2</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</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" title="E30">A12</font></i>: 
not <font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c15">E2</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</font></span>)</span>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c13">x</font>
;<br/><b>take </b>
<font color="Maroon" title="c15">E2</font>
;<br/><b>thus </b><a NAME="E2:29_1_2_2_1_1"/>
( <font color="Maroon" title="c15">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp; ( <font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c15">E2</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</font></span>)</span> implies  ex <font color="Olive" title="b1">g</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <br/>( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">E2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c15">E2</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Olive" title="b1">g</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="tietze.html#E3:29_1_2_2"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="tietze.html#E1:29_1_2_2_1_1"><i><font color="Green" title="E30">A12</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" title="E30">A13</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c15">E2</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</font></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c16">E2a</font> = <font color="Maroon" title="c15">E2</font> as   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">E2a</font> = <font color="Maroon" title="c16">E2a</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">E2b</font> = <font color="Maroon" title="c15">E2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A1</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">E2b</font> = <font color="Maroon" title="c18">E2b</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A1</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/><a NAME="E6:29_1_2_2_1_2"/>
<font color="Maroon" title="c19">E2b</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c15">E2</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></i>;<br/><a NAME="E7:29_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E31">A14</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c19">E2b</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E1:29_1_2_2_1_2"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="tietze.html#T4" target="_self" title="TIETZE:th.4">Th4</a>, <a class="ref" href="tietze.html#T17" target="_self" title="TIETZE:th.17">Th17</a></i>;<br/><b>set </b><font color="Maroon" title="c20">k</font> = <font color="Maroon" title="c9">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c19">E2b</font>;<br/><b>reconsider </b><font color="Maroon" title="c21">k</font> = <font color="Maroon" title="c9">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c19">E2b</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A1</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">f1c</font> = <font color="Maroon" title="c9">f1</font>, <font color="Maroon" title="c23">E2c</font> = <font color="Maroon" title="c19">E2b</font> as   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A1</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">f1c</font> = <font color="Maroon" title="c22">f1c</font>, <font color="Maroon" title="c25">E2c</font> = <font color="Maroon" title="c23">E2c</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A1</font></span>)</span> <b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/><a NAME="E11:29_1_2_2_1_2"/>
<font color="Maroon" title="c24">f1c</font> <a href="partfun3.html#K7" title="PARTFUN3:func.7">-</a> <font color="Maroon" title="c25">E2c</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 ;<br/><a NAME="E12:29_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E32">A15</font></i>: 
<font color="Maroon" title="c21">k</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/><b>set </b><font color="Maroon" title="c26">r</font> = <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</font></span>)</span>;<br/><a NAME="E13:29_1_2_2_1_2"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></i>;<br/><a NAME="E14:29_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/><b>then consider </b><font color="Maroon" title="c27">g</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E16:29_1_2_2_1_2"/><i><font color="Green" title="E33">A16</font></i>: 
<font color="Maroon" title="c27">g</font>, <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">g</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3
 <b>and </b><br/><a NAME="E17:29_1_2_2_1_2"/><i><font color="Green" title="E34">A17</font></i>: 
<font color="Maroon" title="c21">k</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3
 <b>by </b><i><a class="txt" href="tietze.html#E1:29"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="tietze.html#E7:29_1_2_2_1_2"><i><font color="Green" title="E31">A14</font></i></a>, <a class="txt" href="tietze.html#E12:29_1_2_2_1_2"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="tietze.html#T19" target="_self" title="TIETZE:th.19">Th19</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">ga</font> = <font color="Maroon" title="c27">g</font> as   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">ga</font> = <font color="Maroon" title="c28">ga</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/><b>set </b><font color="Maroon" title="c30">y</font> = <font color="Maroon" title="c17">E2a</font> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <font color="Maroon" title="c29">ga</font>;<br/><b>reconsider </b><font color="Maroon" title="c31">y1</font> = <font color="Maroon" title="c17">E2a</font> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <font color="Maroon" title="c29">ga</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><a NAME="E21:29_1_2_2_1_2"/><i><font color="Green" title="E35">A18</font></i>: 
<font color="Maroon" title="c31">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/><a NAME="E22:29_1_2_2_1_2"/>
<font color="Maroon" title="c31">y1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 <b>by </b><i><a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/><a NAME="E23:29_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c17">E2a</font> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <font color="Maroon" title="c29">ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font>
 <b>by </b><i><a class="txt" href="tietze.html#E21:29_1_2_2_1_2"><i><font color="Green" title="E35">A18</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c32">y</font> = <font color="Maroon" title="c17">E2a</font> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <font color="Maroon" title="c29">ga</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">D</font> ;<br/><b>take </b>
<font color="Maroon" title="c32">y</font>
;<br/><b>take </b>
<font color="Maroon" title="c15">E2</font>
;<br/><b>thus </b><a NAME="E25:29_1_2_2_1_2"/>
<font color="Maroon" title="c15">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="tietze.html#E3:29_1_2_2"><i><font color="Green" title="E29">A11</font></i></a></i>;<br/><b>assume </b><a NAME="E26:29_1_2_2_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c15">E2</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</font></span>)</span>
 ;<br/><b>take </b>
<font color="Maroon" title="c27">g</font>
;<br/><b>thus </b><a NAME="E27:29_1_2_2_1_2"/>
<font color="Maroon" title="c32">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">E2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c27">g</font>
 ;<br/><a NAME="E28:29_1_2_2_1_2"/><i><font color="Green" title="E36">A19</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><b>hence </b><a NAME="E29:29_1_2_2_1_2"/>
<font color="Maroon" title="c27">g</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E16:29_1_2_2_1_2"><i><font color="Green" title="E33">A16</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E30:29_1_2_2_1_2"/><i><font color="Green" title="E37">A20</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E31:29_1_2_2_1_2"/><i><font color="Green" title="E38">A21</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">E2b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E32:29_1_2_2_1_2"/><i><font color="Green" title="E39">A22</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><i><font color="Green" title="E40">A23</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c19">E2b</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c27">g</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c19">E2b</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">g</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">E2b</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">g</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="txt" href="tietze.html#E4:29"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="tietze.html#E30:29_1_2_2_1_2"><i><font color="Green" title="E37">A20</font></i></a>, <a class="txt" href="tietze.html#E31:29_1_2_2_1_2"><i><font color="Green" title="E38">A21</font></i></a>, <a class="txt" href="tietze.html#E32:29_1_2_2_1_2"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c19">E2b</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c27">g</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">A</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c19">E2b</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c27">g</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T63" title="RFUNCT_1:th.63">RFUNCT_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">E2b</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c27">g</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T63" title="RFUNCT_1:th.63">RFUNCT_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">E2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c27">g</font>
<b>by </b><i><a class="ref" href="relat_1.html#T101" title="RELAT_1:th.101">RELAT_1:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c15">E2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c27">g</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T63" title="RFUNCT_1:th.63">RFUNCT_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c15">E2</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c27">g</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T63" title="RFUNCT_1:th.63">RFUNCT_1:63</a></i>
;<br/><b>hence </b><a NAME="E35:29_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c15">E2</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E17:29_1_2_2_1_2"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="tietze.html#E28:29_1_2_2_1_2"><i><font color="Green" title="E36">A19</font></i></a>, <a class="txt" href="tietze.html#E33:29_1_2_2_1_2"><i><font color="Green" title="E40">A23</font></i></a>, <a class="ref" href="tietze.html#T18" target="_self" title="TIETZE:th.18">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" title="c12">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c7">D</font><b> such that </b><br/><a NAME="E18:29_1_2"/><i><font color="Green" title="E29">A24</font></i>: 
<font color="Maroon" title="c12">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g0d</font>
 <b>and </b><br/><a NAME="E19:29_1_2"/><i><font color="Green" title="E30">A25</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c12">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c12">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="tietze.html#E16:29_1_2"><i><font color="Green" title="E28">A10</font></i></a>);<br/></i><a NAME="E20:29_1_2"/><i><font color="Green" title="E31">A26</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:29_1_2"/><i><font color="Green" title="E32">A27</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">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" title="c13">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:29_1_2_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c12">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c12">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <b>by </b><i><a class="txt" href="tietze.html#E19:29_1_2"><i><font color="Green" title="E30">A25</font></i></a></i>;<br/><b>hence </b><a NAME="E2:29_1_2_3"/>
<font color="Maroon" title="c12">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c13">F</font> = <font color="Maroon" title="c12">F</font> as    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="tietze.html#E20:29_1_2"><i><font color="Green" title="E31">A26</font></i></a>, <a class="ref" href="seqfunc.html#T1" title="SEQFUNC:th.1">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" title="c14">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:29_1_2_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">n</font>,<font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font>,<font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <b>by </b><i><a class="txt" href="tietze.html#E19:29_1_2"><i><font color="Green" title="E30">A25</font></i></a></i>;<br/><b>hence </b><a NAME="E2:29_1_2_4"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E25:29_1_2"/><b>then </b><i><font color="Green" title="E33">A28</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c13">F</font>
 <b>by </b><i><a class="ref" href="seqfunc.html#D10" title="SEQFUNC:def.10">SEQFUNC:def 10</a></i>;<br/><a NAME="E26:29_1_2"/><i><font color="Green" title="E34">A29</font></i>: 
( 2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 9 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1 is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 9</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> $1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1</span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">A1</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> $1</span>)</span> );<br/><b>consider </b><font color="Maroon" title="c14">E2</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E28:29_1_2"/><i><font color="Green" title="E35">A30</font></i>: 
( <font color="Maroon" title="c14">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( <font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c14">E2</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> implies  ex <font color="Olive" title="b1">g</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <br/>( <font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c14">E2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c14">E2</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Olive" title="b1">g</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="tietze.html#E19:29_1_2"><i><font color="Green" title="E30">A25</font></i></a></i>;<br/><a NAME="E29:29_1_2"/><i><font color="Green" title="E36">A31</font></i>: 
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a>, <a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">g1</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E31:29_1_2"/><i><font color="Green" title="E37">A32</font></i>: 
( <font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c14">E2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c15">g1</font> &amp; <font color="Maroon" title="c15">g1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c14">E2</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c15">g1</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="tietze.html#E12:29_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="tietze.html#E18:29_1_2"><i><font color="Green" title="E29">A24</font></i></a>, <a class="txt" href="tietze.html#E28:29_1_2"><i><font color="Green" title="E35">A30</font></i></a></i>;<br/><a NAME="E32:29_1_2"/><i><font color="Green" title="E38">A33</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>;<br/><i><font color="Green" title="E39">A34</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">g1</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">E2</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/><i><font color="Green" title="E40">A35</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">E2</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c14">E2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">E2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">E2</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">E2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">E2</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">g1</font>
<b>by </b><i><a class="txt" href="tietze.html#E33:29_1_2"><i><font color="Green" title="E39">A34</font></i></a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_2_7"/><i><font color="Green" title="E41">A36</font></i>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">g1</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">E2</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c14">E2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">E2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">E2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">a</font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E34:29_1_2"><i><font color="Green" title="E40">A35</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">E2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">E2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">a</font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E1:29_1_2_7"><i><font color="Green" title="E41">A36</font></i></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>
<br/>.= 
<font color="Maroon" title="c15">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">a</font>

;<br/><br/></div><b>end;</b></div><a NAME="E36:29_1_2"/><b>then </b>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 9</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g1</font> )
 <b>by </b><i><a class="txt" href="tietze.html#E28:29_1_2"><i><font color="Green" title="E35">A30</font></i></a>, <a class="txt" href="tietze.html#E31:29_1_2"><i><font color="Green" title="E37">A32</font></i></a>, <a class="txt" href="tietze.html#E32:29_1_2"><i><font color="Green" title="E38">A33</font></i></a>, <a class="txt" href="tietze.html#E34:29_1_2"><i><font color="Green" title="E40">A35</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><a NAME="E37:29_1_2"/><b>then </b><i><font color="Green" title="E41">A37</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="txt" href="tietze.html#E12:29_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="tietze.html#E18:29_1_2"><i><font color="Green" title="E29">A24</font></i></a>, <a class="txt" href="tietze.html#E29:29_1_2"><i><font color="Green" title="E36">A31</font></i></a>, <a class="txt" href="tietze.html#E31:29_1_2"><i><font color="Green" title="E37">A32</font></i></a>, <a class="ref" href="tietze.html#T24" target="_self" title="TIETZE:th.24">Th24</a></i>;<br/><div><i><font color="Green" title="E42">A38</font></i>: <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" title="c16">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>consider </b><font color="Maroon" title="c17">E2</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E2:29_1_2_8"/><i><font color="Green" title="E43">A39</font></i>: 
( <font color="Maroon" title="c17">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">n</font> &amp; ( <font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c17">E2</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c16">n</font></span>)</span> implies  ex <font color="Olive" title="b1">g</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <br/>( <font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">E2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c17">E2</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Olive" title="b1">g</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="tietze.html#E19:29_1_2"><i><font color="Green" title="E30">A25</font></i></a></i>;<br/><b>assume </b><a NAME="E3:29_1_2_8"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">n</font>]
 ;<br/><a NAME="E4:29_1_2_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">A1</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c16">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">g</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E6:29_1_2_8"/><i><font color="Green" title="E44">A40</font></i>: 
( <font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">E2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c18">g</font> &amp; <font color="Maroon" title="c18">g</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c17">E2</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c18">g</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="tietze.html#E2:29_1_2_8"><i><font color="Green" title="E43">A39</font></i></a>, <a class="ref" href="tietze.html#T24" target="_self" title="TIETZE:th.24">Th24</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">E2'</font> = <font color="Maroon" title="c17">E2</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="tietze.html#E21:29_1_2"><i><font color="Green" title="E32">A27</font></i></a>, <a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">g'</font> = <font color="Maroon" title="c18">g</font> as  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="tietze.html#E21:29_1_2"><i><font color="Green" title="E32">A27</font></i></a>, <a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/><a NAME="E9:29_1_2_8"/><i><font color="Green" title="E45">A41</font></i>: 
<font color="Maroon" title="c19">E2'</font> <a href="partfun3.html#K6" title="PARTFUN3:func.6">+</a> <font color="Maroon" title="c20">g'</font> is  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>
 ;<br/><b>consider </b><font color="Maroon" title="c21">E3</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E11:29_1_2_8"/><i><font color="Green" title="E46">A42</font></i>: 
( <font color="Maroon" title="c21">E3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( <font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c21">E3</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">g</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <br/>( <font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c21">E3</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c21">E3</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Olive" title="b1">g</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="tietze.html#E19:29_1_2"><i><font color="Green" title="E30">A25</font></i></a></i>;<br/><a NAME="E12:29_1_2_8"/><i><font color="Green" title="E47">A43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c17">E2</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c18">g</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E6:29_1_2_8"><i><font color="Green" title="E44">A40</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/><i><font color="Green" title="E48">A44</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A1</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><i><font color="Green" title="E49">A45</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c17">E2</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c18">g</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c17">E2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">g</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">E2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">g</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">g</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">g</font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E13:29_1_2_8"><i><font color="Green" title="E48">A44</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="txt" href="tietze.html#E13:29_1_2_8"><i><font color="Green" title="E48">A44</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><i><font color="Green" title="E50">A46</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">E2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c18">g</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">E2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c18">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">E2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E13:29_1_2_8"><i><font color="Green" title="E48">A44</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">E2</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><i><font color="Green" title="E51">A47</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c17">E2</font></span>)</span> = 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">E2</font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E13:29_1_2_8"><i><font color="Green" title="E48">A44</font></i></a>, <a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><div><i><font color="Green" title="E52">A48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_8_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_2_8_5"/><i><font color="Green" title="E53">A49</font></i>: 
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c17">E2</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c18">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c17">E2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E14:29_1_2_8"><i><font color="Green" title="E49">A45</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">E2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E16:29_1_2_8"><i><font color="Green" title="E51">A47</font></i></a>, <a class="txt" href="tietze.html#E1:29_1_2_8_5"><i><font color="Green" title="E53">A49</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">E2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">E2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c18">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E1:29_1_2_8_5"><i><font color="Green" title="E53">A49</font></i></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">E2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c18">g</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">a</font>
<b>by </b><i><a class="txt" href="tietze.html#E15:29_1_2_8"><i><font color="Green" title="E50">A46</font></i></a>, <a class="txt" href="tietze.html#E1:29_1_2_8_5"><i><font color="Green" title="E53">A49</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E18:29_1_2_8"/><b>then </b><i><font color="Green" title="E53">A50</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">A1</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E6:29_1_2_8"><i><font color="Green" title="E44">A40</font></i></a>, <a class="txt" href="tietze.html#E12:29_1_2_8"><i><font color="Green" title="E47">A43</font></i></a>, <a class="txt" href="tietze.html#E14:29_1_2_8"><i><font color="Green" title="E49">A45</font></i></a>, <a class="txt" href="tietze.html#E15:29_1_2_8"><i><font color="Green" title="E50">A46</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">gx</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E20:29_1_2_8"/><i><font color="Green" title="E54">A51</font></i>: 
( <font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c21">E3</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c22">gx</font> &amp; <font color="Maroon" title="c22">gx</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c21">E3</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c22">gx</font>,<font color="Maroon" title="c2">A</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="tietze.html#E6:29_1_2_8"><i><font color="Green" title="E44">A40</font></i></a>, <a class="txt" href="tietze.html#E11:29_1_2_8"><i><font color="Green" title="E46">A42</font></i></a>, <a class="txt" href="tietze.html#E14:29_1_2_8"><i><font color="Green" title="E49">A45</font></i></a>, <a class="txt" href="tietze.html#E15:29_1_2_8"><i><font color="Green" title="E50">A46</font></i></a>, <a class="txt" href="tietze.html#E17:29_1_2_8"><i><font color="Green" title="E52">A48</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><i><font color="Green" title="E55">A52</font></i>: <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 9</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/><i><font color="Green" title="E56">A53</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">gx</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">E3</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/><i><font color="Green" title="E57">A54</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">gx</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c21">E3</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c21">E3</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">gx</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c21">E3</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">E3</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">gx</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">E3</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">E3</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">gx</font>
<b>by </b><i><a class="txt" href="tietze.html#E22:29_1_2_8"><i><font color="Green" title="E56">A53</font></i></a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_8_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_2_8_9"/><i><font color="Green" title="E58">A55</font></i>: 
<font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">gx</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">gx</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c21">E3</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c21">E3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">gx</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c21">E3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">E3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">a</font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E23:29_1_2_8"><i><font color="Green" title="E57">A54</font></i></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">gx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">E3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">E3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">a</font></span>)</span>
<b>by </b><i><a class="txt" href="tietze.html#E1:29_1_2_8_9"><i><font color="Green" title="E58">A55</font></i></a>, <a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></i>
<br/>.= 
<font color="Maroon" title="c22">gx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">a</font>

;<br/><br/></div><b>end;</b></div><a NAME="E25:29_1_2_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">gx</font>
 <b>by </b><i><a class="txt" href="tietze.html#E11:29_1_2_8"><i><font color="Green" title="E46">A42</font></i></a>, <a class="txt" href="tietze.html#E20:29_1_2_8"><i><font color="Green" title="E54">A51</font></i></a>, <a class="txt" href="tietze.html#E23:29_1_2_8"><i><font color="Green" title="E57">A54</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E26:29_1_2_8"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="tietze.html#E21:29_1_2"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="tietze.html#E6:29_1_2_8"><i><font color="Green" title="E44">A40</font></i></a>, <a class="txt" href="tietze.html#E9:29_1_2_8"><i><font color="Green" title="E45">A41</font></i></a>, <a class="txt" href="tietze.html#E18:29_1_2_8"><i><font color="Green" title="E53">A50</font></i></a>, <a class="txt" href="tietze.html#E20:29_1_2_8"><i><font color="Green" title="E54">A51</font></i></a>, <a class="txt" href="tietze.html#E21:29_1_2_8"><i><font color="Green" title="E55">A52</font></i></a>, <a class="ref" href="tietze.html#T24" target="_self" title="TIETZE:th.24">Th24</a>, <a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/></div><b>end;</b></div><a NAME="E39:29_1_2"/><i><font color="Green" title="E43">A56</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="tietze.html#E37:29_1_2"><i><font color="Green" title="E41">A37</font></i></a>, <a class="txt" href="tietze.html#E38:29_1_2"><i><font color="Green" title="E42">A38</font></i></a>);<br/></i><div><i><font color="Green" title="E44">A57</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" title="c16">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><a NAME="E1:29_1_2_9"/>
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:29_1_2_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 9</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c16">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E39:29_1_2"><i><font color="Green" title="E43">A56</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E41:29_1_2"/><b>then </b><i><font color="Green" title="E45">A58</font></i>: 
<font color="Maroon" title="c13">F</font> <a href="seqfunc.html#R3" title="SEQFUNC:pred.3">is_unif_conv_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="tietze.html#E25:29_1_2"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="tietze.html#E26:29_1_2"><i><font color="Green" title="E34">A29</font></i></a>, <a class="ref" href="tietze.html#T11" target="_self" title="TIETZE:th.11">Th11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">h</font> =  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c13">F</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="tietze.html#E39:29_1_2"><i><font color="Green" title="E43">A56</font></i></a>, <a class="ref" href="tietze.html#T22" target="_self" title="TIETZE:th.22">Th22</a></i>;<br/><div><i><font color="Green" title="E46">A59</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" title="c17">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><a NAME="E1:29_1_2_10"/>
<font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">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" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">A1</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c17">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E39:29_1_2"><i><font color="Green" title="E43">A56</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E44:29_1_2"/><i><font color="Green" title="E47">A60</font></i>: 
<font color="Maroon" title="c4">A1</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c13">F</font>
 <b>by </b><i><a class="txt" href="tietze.html#E25:29_1_2"><i><font color="Green" title="E33">A28</font></i></a>, <a class="ref" href="seqfunc.html#T24" title="SEQFUNC:th.24">SEQFUNC:24</a></i>;<br/><a NAME="E45:29_1_2"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A1</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="valued_0.html#D3" title="VALUED_0:def.3">VALUED_0:def 3</a></i>;<br/><a NAME="E46:29_1_2"/><b>then </b><i><font color="Green" title="E48">A61</font></i>: 
<font color="Maroon" title="c3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">A1</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="txt" href="tietze.html#E4:29"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E47:29_1_2"/>
<font color="Maroon" title="c13">F</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="tietze.html#E41:29_1_2"><i><font color="Green" title="E45">A58</font></i></a>, <a class="ref" href="seqfunc.html#T23" title="SEQFUNC:th.23">SEQFUNC:23</a></i>;<br/><b>then </b><i><font color="Green" title="E49">A62</font></i>: <font color="Maroon" title="c16">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">A1</font> = 
 <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c4">A1</font>
<b>by </b><i><a class="ref" href="seqfunc.html#T25" title="SEQFUNC:th.25">SEQFUNC:25</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="txt" href="tietze.html#E26:29_1_2"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="tietze.html#E43:29_1_2"><i><font color="Green" title="E46">A59</font></i></a>, <a class="txt" href="tietze.html#E44:29_1_2"><i><font color="Green" title="E47">A60</font></i></a>, <a class="txt" href="tietze.html#E46:29_1_2"><i><font color="Green" title="E48">A61</font></i></a>, <a class="ref" href="tietze.html#T13" target="_self" title="TIETZE:th.13">Th13</a></i>
;<br/><a NAME="E49:29_1_2"/>
<font color="Maroon" title="c16">h</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tietze.html#D1" target="_self" title="TIETZE:def.1">TIETZE:def 1</a><br/>

<b>assume </b><a NAME="E1:29_1_2_12"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">h</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c18">z</font> = <font color="Maroon" title="c17">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E3:29_1_2_12"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="tietze.html#E18:29_1_2"><i><font color="Green" title="E29">A24</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:29_1_2_12"/><b>then </b>
( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3 )
 
<b>by </b><i><a class="txt" href="tietze.html#E11:29_1_2"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="tietze.html#E18:29_1_2"><i><font color="Green" title="E29">A24</font></i></a></i>;<br/>
<a NAME="E5:29_1_2_12"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">z</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3
 
<b>by </b><i><a class="ref" href="tietze.html#D1" target="_self" title="TIETZE:def.1">Def1</a></i>;<br/>
<a NAME="E6:29_1_2_12"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3 )
 
<b>by </b><i><a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>
<a NAME="E7:29_1_2_12"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">z</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 9</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 9</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 9</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 9</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c16">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">z</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 9</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 3</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c16">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 9</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K10" title="REAL_1:func.10">/</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" title="E33">A28</font></i></a>, <a class="txt" href="tietze.html#E26:29_1_2"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="tietze.html#E40:29_1_2"><i><font color="Green" title="E44">A57</font></i></a>, <a class="ref" href="tietze.html#T12" target="_self" title="TIETZE:th.12">Th12</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:29_1_2_12"/><b>then </b>
( <font color="Maroon" title="c16">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c16">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:29_1_2_12"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c17">h</font> = <font color="Maroon" title="c16">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="tietze.html#T23" target="_self" title="TIETZE:th.23">Th23</a></i>;<br/><a NAME="E51:29_1_2"/>
 <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E3:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></i>;<br/><a NAME="E52:29_1_2"/><b>then </b>
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E53:29_1_2"/><b>then </b>
<font color="Maroon" title="c17">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T9" title="TOPMETR:th.9">TOPMETR:9</a></i>;<br/><b>hence </b><a NAME="E54:29_1_2"/>
 ex <font color="Olive" title="b1">g</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> st <font color="Olive" title="b1">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="tietze.html#E48:29_1_2"><i><font color="Green" title="E49">A62</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
