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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font> is   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</a></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">B</font> =  <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">B</font> =  <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font> as   <a href="lopban_2.html#NM2" title="LOPBAN_2:NM.2">Normed_Algebra</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T22" target="_self" title="C0SP1:th.22">Th23</a></span>;<br/>
<a NAME="E2:69"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">ONE</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E4:69"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c4">ONE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E5:69"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<font color="Maroon" title="c4">ONE</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/>
<a NAME="E6:69"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K7" title="C0SP1:func.7">R_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
;<br/>
<a NAME="E7:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A3</font></span>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T16" target="_self" title="C0SP1:th.16">Th16</a></span>;<br/>
<a NAME="E8:69"/>
 for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c4">ONE</font> iff <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c5">t</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c4">ONE</font> iff <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<a NAME="E1:69_1"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c4">ONE</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:69_1_1"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c4">ONE</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:69_1_1"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E7:69"><span class="lab"><font color="Green" title="E51">A3</font></span></a></span>;<br/><a NAME="E4:69_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E5:69_1_1"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="complex1.html#T48" title="COMPLEX1:th.48">COMPLEX1:48</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:69_1"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c4">ONE</font></span><br/>

<a NAME="E4:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="ref" href="complex1.html#T48" title="COMPLEX1:th.48">COMPLEX1:48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:69_1"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c4">ONE</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E7:69"><span class="lab"><font color="Green" title="E51">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:69"/><span class="kw">then </span>
 <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c4">ONE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default">1</span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E10:69"/><span class="kw">then </span>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c4">ONE</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T9" title="SEQ_4:th.9">SEQ_4:9</a></span>;<br/>
<a NAME="E11:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A6</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">B</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="c0sp1.html#T20" target="_self" title="C0SP1:th.20">Th21</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E53">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> holds  <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">g</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c7">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">g</font></span>)</span></span><br/><a NAME="E1:69_2"/>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E3:69_2"/><span class="lab"><font color="Green" title="E54">A8</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">and </span><br/><a NAME="E4:69_2"/>
<font color="Maroon" title="c8">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E5:69_2"/>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">g1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E7:69_2"/><span class="lab"><font color="Green" title="E55">A9</font></span>: 
<font color="Maroon" title="c9">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">and </span><br/><a NAME="E8:69_2"/>
<font color="Maroon" title="c9">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E9:69_2"/>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">h3</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E11:69_2"/><span class="lab"><font color="Green" title="E56">A10</font></span>: 
<font color="Maroon" title="c10">h3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:69_2"/>
<font color="Maroon" title="c10">h3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E13:69_2"/>
<font color="Maroon" title="c6">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">h2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E15:69_2"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
<font color="Maroon" title="c11">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">and </span><br/><a NAME="E16:69_2"/>
<font color="Maroon" title="c11">h2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E17:69_2"/>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">h1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E19:69_2"/><span class="lab"><font color="Green" title="E58">A12</font></span>: 
<font color="Maroon" title="c12">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">and </span><br/><a NAME="E20:69_2"/>
<font color="Maroon" title="c12">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">h3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><br/><a NAME="E1:69_2_1"/>
<font color="Maroon" title="c10">h3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:69_2"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="txt" href="#E11:69_2"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="ref" href="c0sp1.html#T30" target="_self" title="C0SP1:th.30">Th31</a></span>;<br/><a NAME="E2:69_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">h3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E7:69_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E15:69_2"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="ref" href="c0sp1.html#T31" target="_self" title="C0SP1:th.31">Th32</a></span>;<br/><a NAME="E3:69_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">h3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E4:69_2_1"/>
<font color="Maroon" title="c10">h3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:69_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E19:69_2"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="ref" href="c0sp1.html#T30" target="_self" title="C0SP1:th.30">Th31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E22:69_2"/>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E19:69_2"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="txt" href="#E11:69_2"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="ref" href="c0sp1.html#T31" target="_self" title="C0SP1:th.31">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E54">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">g</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">g</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><a NAME="E1:69_3"/>
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E3:69_3"/><span class="lab"><font color="Green" title="E55">A14</font></span>: 
<font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E4:69_3"/><span class="lab"><font color="Green" title="E56">A15</font></span>: 
<font color="Maroon" title="c7">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E5:69_3"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">g1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E7:69_3"/><span class="lab"><font color="Green" title="E57">A16</font></span>: 
<font color="Maroon" title="c8">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">and </span><br/><a NAME="E8:69_3"/><span class="lab"><font color="Green" title="E58">A17</font></span>: 
<font color="Maroon" title="c8">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E9:69_3"/><span class="lab"><font color="Green" title="E59">A18</font></span>: 
(  not  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c8">g1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c8">g1</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:69_3"><span class="lab"><font color="Green" title="E58">A17</font></span></a>, <a class="ref" href="c0sp1.html#T17" target="_self" title="C0SP1:th.17">Th17</a></span>;<br/><a NAME="E10:69_3"/>
<font color="Maroon" title="c5">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">h1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E12:69_3"/><span class="lab"><font color="Green" title="E60">A19</font></span>: 
<font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">and </span><br/><a NAME="E13:69_3"/><span class="lab"><font color="Green" title="E61">A20</font></span>: 
<font color="Maroon" title="c9">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E14:69_3"/><span class="lab"><font color="Green" title="E62">A21</font></span>: 
(  not  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_3"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="ref" href="c0sp1.html#T17" target="_self" title="C0SP1:th.17">Th17</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">s</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c9">h1</font> implies <font color="Maroon" title="c10">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c8">g1</font></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:69_3_1"/>
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c9">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c8">g1</font></span>)</span></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:69_3_1"/><span class="lab"><font color="Green" title="E63">A22</font></span>: 
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span>
 ;<br/><a NAME="E4:69_3_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c8">g1</font>
 ;<br/><a NAME="E5:69_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A23</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c8">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69_3"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>;<br/><a NAME="E6:69_3_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font>
 ;<br/><a NAME="E7:69_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A24</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:69_3"><span class="lab"><font color="Green" title="E62">A21</font></span></a>, <a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>;<br/><a NAME="E8:69_3_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/><a NAME="E9:69_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A25</font></span>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c8">g1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_3_1"><span class="lab"><font color="Green" title="E64">A23</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E10:69_3_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/><a NAME="E11:69_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A26</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:69_3_1"><span class="lab"><font color="Green" title="E65">A24</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E12:69_3_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_3"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="txt" href="#E7:69_3"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E12:69_3"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="ref" href="c0sp1.html#T31" target="_self" title="C0SP1:th.31">Th32</a></span>;<br/><a NAME="E13:69_3_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" title="COMPLEX1:th.65">COMPLEX1:65</a></span>;<br/><span class="kw">hence </span><a NAME="E14:69_3_1"/>
<font color="Maroon" title="c10">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c8">g1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_3_1"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="txt" href="#E11:69_3_1"><span class="lab"><font color="Green" title="E67">A26</font></span></a>, <a class="txt" href="#E9:69_3_1"><span class="lab"><font color="Green" title="E66">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E16:69_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A27</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c9">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c8">g1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T45" title="SEQ_4:th.45">SEQ_4:45</a></span>;<br/><a NAME="E17:69_3"/><span class="lab"><font color="Green" title="E64">A28</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">g</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c8">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:69_3"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E8:69_3"><span class="lab"><font color="Green" title="E58">A17</font></span></a>, <a class="ref" href="c0sp1.html#T20" target="_self" title="C0SP1:th.20">Th21</a></span>;<br/><a NAME="E18:69_3"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_3"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="txt" href="#E4:69_3"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="ref" href="c0sp1.html#T20" target="_self" title="C0SP1:th.20">Th21</a></span>;<br/><span class="kw">hence </span><a NAME="E19:69_3"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">g</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">g</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:69_3"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="txt" href="#E13:69_3"><span class="lab"><font color="Green" title="E61">A20</font></span></a>, <a class="txt" href="#E17:69_3"><span class="lab"><font color="Green" title="E64">A28</font></span></a>, <a class="txt" href="#E16:69_3"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="ref" href="c0sp1.html#T20" target="_self" title="C0SP1:th.20">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E55">A29</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c7">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span></span><br/><a NAME="E1:69_4"/>
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E3:69_4"/><span class="lab"><font color="Green" title="E56">A30</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E4:69_4"/>
<font color="Maroon" title="c8">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E5:69_4"/>
<font color="Maroon" title="c7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">h1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E7:69_4"/><span class="lab"><font color="Green" title="E57">A31</font></span>: 
<font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font>
 <span class="kw">and </span><br/><a NAME="E8:69_4"/>
<font color="Maroon" title="c9">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E9:69_4"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">g1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E11:69_4"/><span class="lab"><font color="Green" title="E58">A32</font></span>: 
<font color="Maroon" title="c10">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">and </span><br/><a NAME="E12:69_4"/>
<font color="Maroon" title="c10">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E13:69_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">F1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E15:69_4"/><span class="lab"><font color="Green" title="E59">A33</font></span>: 
<font color="Maroon" title="c11">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E16:69_4"/>
<font color="Maroon" title="c11">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E17:69_4"/>
<font color="Maroon" title="c7">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">hf1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E19:69_4"/><span class="lab"><font color="Green" title="E60">A34</font></span>: 
<font color="Maroon" title="c12">hf1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E20:69_4"/>
<font color="Maroon" title="c12">hf1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E21:69_4"/>
<font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">gf1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E23:69_4"/><span class="lab"><font color="Green" title="E61">A35</font></span>: 
<font color="Maroon" title="c13">gf1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E24:69_4"/>
<font color="Maroon" title="c13">gf1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><a NAME="E25:69_4"/>
<font color="Maroon" title="c6">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">gPh1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E27:69_4"/><span class="lab"><font color="Green" title="E62">A36</font></span>: 
<font color="Maroon" title="c14">gPh1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font>
 <span class="kw">and </span><br/><a NAME="E28:69_4"/>
<font color="Maroon" title="c14">gPh1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">gf1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">hf1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span><br/><a NAME="E1:69_4_1"/>
<font color="Maroon" title="c11">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">gPh1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_4"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="txt" href="#E27:69_4"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="txt" href="#E15:69_4"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="ref" href="c0sp1.html#T31" target="_self" title="C0SP1:th.31">Th32</a></span>;<br/><a NAME="E2:69_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</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="c9">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:69_4"><span class="lab"><font color="Green" title="E58">A32</font></span></a>, <a class="txt" href="#E7:69_4"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="txt" href="#E27:69_4"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="c0sp1.html#T29" target="_self" title="C0SP1:th.29">Th30</a></span>;<br/><a NAME="E3:69_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span></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"><font color="Maroon" title="c9">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span>)</span>
 ;<br/><a NAME="E4:69_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">gf1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</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"><font color="Maroon" title="c9">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_4"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="txt" href="#E11:69_4"><span class="lab"><font color="Green" title="E58">A32</font></span></a>, <a class="txt" href="#E23:69_4"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="ref" href="c0sp1.html#T31" target="_self" title="C0SP1:th.31">Th32</a></span>;<br/><span class="kw">hence </span><a NAME="E5:69_4_1"/>
<font color="Maroon" title="c11">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">gf1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">hf1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_4"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="txt" href="#E7:69_4"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="txt" href="#E19:69_4"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="ref" href="c0sp1.html#T31" target="_self" title="C0SP1:th.31">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E30:69_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:69_4"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="txt" href="#E19:69_4"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E15:69_4"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="ref" href="c0sp1.html#T29" target="_self" title="C0SP1:th.29">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E15:69"/>
 for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> holds  <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="c0sp1.html#E42"><span class="lab"><font color="Green" title="E33">Lm4</font></span></a></span>;<br/>
<a NAME="E16:69"/><span class="kw">then </span>
<font color="Maroon" title="c3">B</font> is   <a href="lopban_1.html#V3" title="LOPBAN_1:attr.3">complete</a>   <a href="vectsp_1.html#V2" title="VECTSP_1:attr.2">left-distributive</a>   <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a>   <a href="lopban_2.html#V2" title="LOPBAN_2:attr.2">Banach_Algebra-like_1</a>   <a href="lopban_2.html#V3" title="LOPBAN_2:attr.3">Banach_Algebra-like_2</a>   <a href="lopban_2.html#V4" title="LOPBAN_2:attr.4">Banach_Algebra-like_3</a>  <a href="lopban_2.html#NM2" title="LOPBAN_2:NM.2">Normed_Algebra</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:69"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="txt" href="#E11:69"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="#E12:69"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E14:69"><span class="lab"><font color="Green" title="E55">A29</font></span></a>, <a class="ref" href="lopban_2.html#D9" title="LOPBAN_2:def.9">LOPBAN_2:def 9</a>, <a class="ref" href="lopban_2.html#D10" title="LOPBAN_2:def.10">LOPBAN_2:def 10</a>, <a class="ref" href="lopban_2.html#D11" title="LOPBAN_2:def.11">LOPBAN_2:def 11</a>, <a class="ref" href="vectsp_1.html#D3" title="VECTSP_1:def.3">VECTSP_1:def 3</a>, <a class="ref" href="vectsp_1.html#D8" title="VECTSP_1:def.8">VECTSP_1:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:69"/>
 <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font> is   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
