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<div class="add">

<span class="kw">set </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="c2">X</font>;<br/>
<span class="kw">thus </span><a NAME="E1:68_1_1"/>
 <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font> is  <a href="lopban_2.html#V2" title="LOPBAN_2:attr.2">Banach_Algebra-like_1</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c2">X</font> is  <a href="lopban_2.html#V3" title="LOPBAN_2:attr.3">Banach_Algebra-like_2</a>  &amp;  <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font> is  <a href="lopban_2.html#V4" title="LOPBAN_2:attr.4">Banach_Algebra-like_3</a>  &amp;  <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font> is  <a href="vectsp_1.html#V2" title="VECTSP_1:attr.2">left-distributive</a>  &amp;  <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font> is  <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lopban_2.html#D9" onmouseover="rs('lopban_2/D9')" onmouseout="rh()">LOPBAN_2:def 9</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/>

<a NAME="E1:68_1_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">X</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E3:68_1_1_1"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E4:68_1_1_1"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c6">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E5:68_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">X</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">g1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E7:68_1_1_1"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c7">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">and </span><br/><a NAME="E8:68_1_1_1"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c7">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E9:68_1_1_1"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
(  not  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c7">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="c7">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="#E8:68_1_1_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="c0sp1.html#T17" target="_self" onmouseover="rs('c0sp1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E10:68_1_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">X</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">h1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E12:68_1_1_1"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
<font color="Maroon" title="c8">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">and </span><br/><a NAME="E13:68_1_1_1"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
<font color="Maroon" title="c8">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E14:68_1_1_1"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
(  not  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c6">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="c6">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="#E4:68_1_1_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="c0sp1.html#T17" target="_self" onmouseover="rs('c0sp1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <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="c8">h1</font> holds <br/><font color="Olive" title="b1">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#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c6">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">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">g1</font></span>)</span></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">s</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">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="c8">h1</font> implies <font color="Maroon" title="c9">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#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c6">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">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">g1</font></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:68_1_1_1_1"/>
<font color="Maroon" title="c9">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="c8">h1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">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#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c6">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">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">g1</font></span>)</span></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><br/><a NAME="E3:68_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 ;<br/><a NAME="E4:68_1_1_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></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">g1</font>
 ;<br/><a NAME="E5:68_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A10</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">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">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68_1_1_1"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="seq_4.html#D1" onmouseover="rs('seq_4/D1')" onmouseout="rh()">SEQ_4:def 1</a></span>;<br/><a NAME="E6:68_1_1_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></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="c6">f1</font>
 ;<br/><a NAME="E7:68_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A11</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c6">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:68_1_1_1"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="seq_4.html#D1" onmouseover="rs('seq_4/D1')" onmouseout="rh()">SEQ_4:def 1</a></span>;<br/><a NAME="E8:68_1_1_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a></span>;<br/><a NAME="E9:68_1_1_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c6">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68_1_1_1_1"><span class="lab"><font color="Green" title="E53">A11</font></span></a></span>;<br/><a NAME="E10:68_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A12</font></span>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c6">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></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#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c6">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">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">g1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68_1_1_1_1"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E11:68_1_1_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" onmouseover="rs('complex1/T46')" onmouseout="rh()">COMPLEX1:46</a></span>;<br/><a NAME="E12:68_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A13</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></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#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c6">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68_1_1_1_1"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E13:68_1_1_1_1"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_1_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E7:68_1_1_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E12:68_1_1_1"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="c0sp1.html#T31" target="_self" onmouseover="rs('c0sp1/T31')" onmouseout="rh()">Th31</a></span>;<br/><a NAME="E14:68_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a></span>;<br/><span class="kw">hence </span><a NAME="E15:68_1_1_1_1"/>
<font color="Maroon" title="c9">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#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c6">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">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">g1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_1_1_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E12:68_1_1_1_1"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E10:68_1_1_1_1"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E16:68_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A14</font></span>: 
 <a href="seq_4.html#K2" title="SEQ_4:func.2">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">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#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c6">f1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">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">g1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T45" onmouseover="rs('seq_4/T45')" onmouseout="rh()">SEQ_4:45</a></span>;<br/>
<a NAME="E17:68_1_1_1"/><span class="lab"><font color="Green" title="E52">A15</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">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">g1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68_1_1_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E8:68_1_1_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="c0sp1.html#T20" target="_self" onmouseover="rs('c0sp1/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E18:68_1_1_1"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c6">f1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_1_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E4:68_1_1_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="c0sp1.html#T20" target="_self" onmouseover="rs('c0sp1/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:68_1_1_1"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68_1_1_1"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E13:68_1_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E17:68_1_1_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E16:68_1_1_1"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="c0sp1.html#T20" target="_self" onmouseover="rs('c0sp1/T20')" onmouseout="rh()">Th20</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:68_1_1"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E4:68_1_1"/><span class="lab"><font color="Green" title="E43">A16</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> <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:68_1_1"/><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon" title="c4">ONE</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E6:68_1_1"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font></span>)</span> <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="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E7:68_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A18</font></span>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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" onmouseover="rs('c0sp1/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<a NAME="E8:68_1_1"/>
 for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_1_2"><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="c2">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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E1:68_1_1_2_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c2">X</font><span class="kw"> such that </span><br/><a NAME="E3:68_1_1_2_1"/><span class="lab"><font color="Green" title="E46">A20</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">ONE</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 ;<br/><a NAME="E4:68_1_1_2_1"/><span class="lab"><font color="Green" title="E47">A21</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">t</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E7:68_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E3:68_1_1_2_1"><span class="lab"><font color="Green" title="E46">A20</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E5:68_1_1_2_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="#E4:68_1_1_2_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="ref" href="complex1.html#T48" onmouseover="rs('complex1/T48')" onmouseout="rh()">COMPLEX1:48</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:68_1_1_2"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:68_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T48" onmouseover="rs('complex1/T48')" onmouseout="rh()">COMPLEX1:48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:68_1_1_2"/>
<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="#E4:68_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E7:68_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E9:68_1_1"/><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" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E10:68_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K2" title="SEQ_4:func.2">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" onmouseover="rs('seq_4/T9')" onmouseout="rh()">SEQ_4:9</a></span>;<br/>
<a NAME="E11:68_1_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</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="#E4:68_1_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E5:68_1_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="c0sp1.html#T20" target="_self" onmouseover="rs('c0sp1/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:68_1_1"/>
 <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font> is  <a href="lopban_2.html#V3" title="LOPBAN_2:attr.3">Banach_Algebra-like_2</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c2">X</font> is  <a href="lopban_2.html#V4" title="LOPBAN_2:attr.4">Banach_Algebra-like_3</a>  &amp;  <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font> is  <a href="vectsp_1.html#V2" title="VECTSP_1:attr.2">left-distributive</a>  &amp;  <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font> is  <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E13:68_1_1"/>
 <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font> is  <a href="lopban_2.html#V4" title="LOPBAN_2:attr.4">Banach_Algebra-like_3</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c2">X</font> is  <a href="vectsp_1.html#V2" title="VECTSP_1:attr.2">left-distributive</a>  &amp;  <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font> is  <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lopban_2.html#D11" onmouseover="rs('lopban_2/D11')" onmouseout="rh()">LOPBAN_2:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font></span>)</span> 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">b<sub>1</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></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 <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E1:68_1_1_3"/>
<font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E3:68_1_1_3"/><span class="lab"><font color="Green" title="E46">A22</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="E4:68_1_1_3"/>
<font color="Maroon" title="c8">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E5:68_1_1_3"/>
<font color="Maroon" title="c7">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E7:68_1_1_3"/><span class="lab"><font color="Green" title="E47">A23</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="E8:68_1_1_3"/>
<font color="Maroon" title="c9">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E9:68_1_1_3"/>
<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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E11:68_1_1_3"/><span class="lab"><font color="Green" title="E48">A24</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="E12:68_1_1_3"/>
<font color="Maroon" title="c10">h3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E13:68_1_1_3"/>
<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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E15:68_1_1_3"/><span class="lab"><font color="Green" title="E49">A25</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="E16:68_1_1_3"/>
<font color="Maroon" title="c11">h2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E17:68_1_1_3"/>
<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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E19:68_1_1_3"/><span class="lab"><font color="Green" title="E50">A26</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="E20:68_1_1_3"/>
<font color="Maroon" title="c12">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_1_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> holds  <font color="Maroon" title="c10">h3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></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="c2">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><br/><a NAME="E1:68_1_1_3_1"/>
<font color="Maroon" title="c10">h3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:68_1_1_3"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="txt" href="#E11:68_1_1_3"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="ref" href="c0sp1.html#T30" target="_self" onmouseover="rs('c0sp1/T30')" onmouseout="rh()">Th30</a></span>;<br/><a NAME="E2:68_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">h3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_1_3"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E7:68_1_1_3"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E15:68_1_1_3"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="ref" href="c0sp1.html#T31" target="_self" onmouseover="rs('c0sp1/T31')" onmouseout="rh()">Th31</a></span>;<br/><a NAME="E3:68_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">h3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E4:68_1_1_3_1"/>
<font color="Maroon" title="c10">h3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68_1_1_3"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E19:68_1_1_3"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="c0sp1.html#T30" target="_self" onmouseover="rs('c0sp1/T30')" onmouseout="rh()">Th30</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E22:68_1_1_3"/>
<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="#E3:68_1_1_3"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E19:68_1_1_3"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E11:68_1_1_3"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="ref" href="c0sp1.html#T31" target="_self" onmouseover="rs('c0sp1/T31')" onmouseout="rh()">Th31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">thus </span><a NAME="E14:68_1_1"/>
 <a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font> is  <a href="vectsp_1.html#V2" title="VECTSP_1:attr.2">left-distributive</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c2">X</font> is  <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_1_4"><span class="kw">proof </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 <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D3" onmouseover="rs('vectsp_1/D3')" onmouseout="rh()">VECTSP_1:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="E1:68_1_1_4"/>
<font color="Maroon" title="c5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E3:68_1_1_4"/><span class="lab"><font color="Green" title="E46">A27</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="E4:68_1_1_4"/>
<font color="Maroon" title="c8">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E5:68_1_1_4"/>
<font color="Maroon" title="c7">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E7:68_1_1_4"/><span class="lab"><font color="Green" title="E47">A28</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="E8:68_1_1_4"/>
<font color="Maroon" title="c9">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E9:68_1_1_4"/>
<font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E11:68_1_1_4"/><span class="lab"><font color="Green" title="E48">A29</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="E12:68_1_1_4"/>
<font color="Maroon" title="c10">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E13:68_1_1_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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E15:68_1_1_4"/><span class="lab"><font color="Green" title="E49">A30</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="E16:68_1_1_4"/>
<font color="Maroon" title="c11">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E17:68_1_1_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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E19:68_1_1_4"/><span class="lab"><font color="Green" title="E50">A31</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="E20:68_1_1_4"/>
<font color="Maroon" title="c12">hf1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E21:68_1_1_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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E23:68_1_1_4"/><span class="lab"><font color="Green" title="E51">A32</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="E24:68_1_1_4"/>
<font color="Maroon" title="c13">gf1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<a NAME="E25:68_1_1_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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="c0sp1.html#K6" title="C0SP1:func.6">BoundedFunctions</a> <font color="Maroon" title="c2">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="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E27:68_1_1_4"/><span class="lab"><font color="Green" title="E52">A33</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="E28:68_1_1_4"/>
<font color="Maroon" title="c14">gPh1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_1_4_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> holds  <font color="Maroon" title="c11">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">hf1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></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="c2">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">hf1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span><br/><a NAME="E1:68_1_1_4_1"/>
<font color="Maroon" title="c11">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_1_4"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E27:68_1_1_4"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E15:68_1_1_4"><span class="lab"><font color="Green" title="E49">A30</font></span></a>, <a class="ref" href="c0sp1.html#T31" target="_self" onmouseover="rs('c0sp1/T31')" onmouseout="rh()">Th31</a></span>;<br/><a NAME="E2:68_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68_1_1_4"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="txt" href="#E7:68_1_1_4"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="txt" href="#E27:68_1_1_4"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="ref" href="c0sp1.html#T29" target="_self" onmouseover="rs('c0sp1/T29')" onmouseout="rh()">Th29</a></span>;<br/><a NAME="E3:68_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span>)</span>
 ;<br/><a NAME="E4:68_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_1_4"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E11:68_1_1_4"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="txt" href="#E23:68_1_1_4"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="ref" href="c0sp1.html#T31" target="_self" onmouseover="rs('c0sp1/T31')" onmouseout="rh()">Th31</a></span>;<br/><span class="kw">hence </span><a NAME="E5:68_1_1_4_1"/>
<font color="Maroon" title="c11">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">hf1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_1_4"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E7:68_1_1_4"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="txt" href="#E19:68_1_1_4"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="ref" href="c0sp1.html#T31" target="_self" onmouseover="rs('c0sp1/T31')" onmouseout="rh()">Th31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E30:68_1_1_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="#E23:68_1_1_4"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E19:68_1_1_4"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="txt" href="#E15:68_1_1_4"><span class="lab"><font color="Green" title="E49">A30</font></span></a>, <a class="ref" href="c0sp1.html#T29" target="_self" onmouseover="rs('c0sp1/T29')" onmouseout="rh()">Th29</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">thus </span><a NAME="E15:68_1_1"/>
 for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c2">X</font></span>)</span></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="E27">Lm3</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D8" onmouseover="rs('vectsp_1/D8')" onmouseout="rh()">VECTSP_1:def 8</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
