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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</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="lopban10.html#K17" title="LOPBAN10:func.17">BoundedMultilinearOperatorsNorm</a> (<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lopban_1.html#K15" title="LOPBAN_1:func.15">BoundedLinearOperatorsNorm</a> (<span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Y</font>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">n1</font> =  <a href="lopban10.html#K17" title="LOPBAN10:func.17">BoundedMultilinearOperatorsNorm</a> (<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">n2</font> =  <a href="lopban_1.html#K15" title="LOPBAN_1:func.15">BoundedLinearOperatorsNorm</a> (<span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Y</font>);<br/>
<a NAME="E1:35"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <a href="lopban10.html#K12" title="LOPBAN10:func.12">BoundedMultilinearOperators</a> (<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> (<span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T22" target="_self" onmouseover="rs('lopban14/T22')" onmouseout="rh()">Th23</a></span>;<br/>
<a NAME="E2:35"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <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="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#T38" onmouseover="rs('finseq_1/T38')" onmouseout="rh()">FINSEQ_1:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">i</font> = 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</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="E4:35"/>
 for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K12" title="LOPBAN10:func.12">BoundedMultilinearOperators</a> (<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font>) holds <br/><span class="p1">(<span class="default"><a href="lopban10.html#K17" title="LOPBAN10:func.17">BoundedMultilinearOperatorsNorm</a> (<span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K15" title="LOPBAN_1:func.15">BoundedLinearOperatorsNorm</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">f</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">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K12" title="LOPBAN10:func.12">BoundedMultilinearOperators</a> (<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font>) implies <span class="p1">(<span class="default"><a href="lopban10.html#K17" title="LOPBAN10:func.17">BoundedMultilinearOperatorsNorm</a> (<span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K15" title="LOPBAN_1:func.15">BoundedLinearOperatorsNorm</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K12" title="LOPBAN10:func.12">BoundedMultilinearOperators</a> (<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</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"> <span class="p1">(<span class="default"><a href="lopban10.html#K17" title="LOPBAN10:func.17">BoundedMultilinearOperatorsNorm</a> (<span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K15" title="LOPBAN_1:func.15">BoundedLinearOperatorsNorm</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c6">f</font> as   <a href="lopban10.html#V3" title="LOPBAN10:attr.3">Lipschitzian</a>  <a href="lopban10.html#NM2" title="LOPBAN10:NM.2">MultilinearOperator</a> of <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="lopban10.html#D11" onmouseover="rs('lopban10/D11')" onmouseout="rh()">LOPBAN10:def 11</a></span>;<br/>
<a NAME="E3:35_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
 <a href="lopban10.html#K15" title="LOPBAN10:func.15">modetrans</a> (<font color="Maroon" title="c6">f</font>,<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">f1</font>
 
;<br/>
<a NAME="E4:35_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<span class="p1">(<span class="default"><a href="lopban10.html#K17" title="LOPBAN10:func.17">BoundedMultilinearOperatorsNorm</a> (<span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font> <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="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c7">f1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="lopban10.html#D15" onmouseover="rs('lopban10/D15')" onmouseout="rh()">LOPBAN10:def 15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">f2</font> = <font color="Maroon" title="c6">f</font> as   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">Lipschitzian</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="lopban_1.html#D9" onmouseover="rs('lopban_1/D9')" onmouseout="rh()">LOPBAN_1:def 9</a></span>;<br/>
<a NAME="E6:35_1"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
 <a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> (<font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="lopban_1.html#D11" onmouseover="rs('lopban_1/D11')" onmouseout="rh()">LOPBAN_1:def 11</a></span>;<br/>
<a NAME="E7:35_1"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<span class="p1">(<span class="default"><a href="lopban_1.html#K15" title="LOPBAN_1:func.15">BoundedLinearOperatorsNorm</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font> <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="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c8">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E6:35_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="lopban_1.html#D13" onmouseover="rs('lopban_1/D13')" onmouseout="rh()">LOPBAN_1:def 13</a></span>;<br/>
<a NAME="E8:35_1"/>
 for <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c7">f1</font> iff <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c8">f2</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="35_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">n</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="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c7">f1</font> iff <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c8">f2</font> )</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="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c8">f2</font> implies <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c7">f1</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1_1_1"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c7">f1</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">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c8">f2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:35_1_1_1"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c7">f1</font> <a href="lopban10.html#K9" title="LOPBAN10:func.9">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> holds  <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="c10">x</font> <a href="lopban10.html#K10" title="LOPBAN10:func.10">.</a> <font color="Olive" title="b1">i</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> 1 ) )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c11">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:35_1_1_1"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T11" target="_self" onmouseover="rs('lopban14/T11')" onmouseout="rh()">Th12</a></span>;<br/><a NAME="E6:35_1_1_1"/><span class="lab"><font color="Green" title="E31">A11</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c10">x</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> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c11">x1</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="#E5:35_1_1_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="ref" href="lopban14.html#T11" target="_self" onmouseover="rs('lopban14/T11')" onmouseout="rh()">Th12</a></span>;<br/><a NAME="E7:35_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="c10">x</font> <a href="lopban10.html#K10" title="LOPBAN10:func.10">.</a> <font color="Maroon" title="c5">i</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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>;<br/><a NAME="E8:35_1_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"><font color="Maroon" title="c11">x1</font></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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:35_1_1_1"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c8">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E6:35_1_1_1"><span class="lab"><font color="Green" title="E31">A11</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>

<span class="kw">assume </span><a NAME="E2:35_1_1"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c8">f2</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">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c7">f1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">s</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:35_1_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c8">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">s</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c10">s</font></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> 1 )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">s1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E6:35_1_1"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">s1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T11" target="_self" onmouseover="rs('lopban14/T11')" onmouseout="rh()">Th12</a></span>;<br/>
<a NAME="E7:35_1_1"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c10">s</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> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c11">s1</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="#E6:35_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="lopban14.html#T11" target="_self" onmouseover="rs('lopban14/T11')" onmouseout="rh()">Th12</a></span>;<br/>
<a NAME="E8:35_1_1"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> holds  <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="c10">s</font> <a href="lopban10.html#K10" title="LOPBAN10:func.10">.</a> <font color="Olive" title="b1">i</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> 1
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="35_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</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"> <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="c10">s</font> <a href="lopban10.html#K10" title="LOPBAN10:func.10">.</a> <font color="Maroon" title="c12">i</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> 1</span><br/>

<a NAME="E1:35_1_1_2"/><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:35_1_1_2"/>
<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="c10">s</font> <a href="lopban10.html#K10" title="LOPBAN10:func.10">.</a> <font color="Maroon" title="c12">i</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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E7:35_1_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E1:35_1_1_2"><span class="lab"><font color="Green" title="E32">A15</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>
<span class="kw">hence </span><a NAME="E9:35_1_1"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban10.html#K16" title="LOPBAN10:func.16">PreNorms</a> <font color="Maroon" title="c7">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1"><span class="lab"><font color="Green" title="E29">A12</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>
<span class="kw">hence </span><a NAME="E9:35_1"/>
<span class="p1">(<span class="default"><a href="lopban10.html#K17" title="LOPBAN10:func.17">BoundedMultilinearOperatorsNorm</a> (<span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K15" title="LOPBAN_1:func.15">BoundedLinearOperatorsNorm</a> (<span class="p2">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E7:35_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI: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>
<span class="kw">hence </span><a NAME="E5:35"/>
 <a href="lopban10.html#K17" title="LOPBAN10:func.17">BoundedMultilinearOperatorsNorm</a> (<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c2">Y</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lopban_1.html#K15" title="LOPBAN_1:func.15">BoundedLinearOperatorsNorm</a> (<span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Y</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban14.html#T22" target="_self" onmouseover="rs('lopban14/T22')" onmouseout="rh()">Th23</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>
