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<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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span><br/>  for <font color="Olive" title="b2">g</font> being   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">bounded</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/> for <font color="Olive" title="b3">t</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b3">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">bounded</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> holds <br/> for <font color="Olive" title="b2">t</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b2">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">g</font> be   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">bounded</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> implies  for <font color="Olive" title="b1">t</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:60"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/>

<a NAME="E2:60"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
(  not  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c4">g</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c4">g</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="ref" href="lopban_1.html#T27" target="_self" title="LOPBAN_1:th.27">Th32</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">t</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:60_1_1_1"/>
( <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c5">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:60_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><div class="add"><a NAME="E2:60_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font> = 
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_1_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D5" target="_self" title="LOPBAN_1:def.5">Def6</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:60_1_1_1_1"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_1_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:60_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c6">t1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">t</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:60_1_1_1_2"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_1_2"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="normsp_0.html#D5" title="NORMSP_0:def.5">NORMSP_0:def 5</a></span>;<br/><span class="lab"><font color="Green" title="E41">A7</font></span>:  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 

;<br/><span class="lab"><font color="Green" title="E42">A8</font></span>: <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> = 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_1_1_2"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="ref" href="normsp_1.html#D1" title="NORMSP_1:def.1">NORMSP_1:def 1</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t1</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D5" target="_self" title="LOPBAN_1:def.5">Def6</a></span>
;<br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">t1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D1" title="NORMSP_1:def.1">NORMSP_1:def 1</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_1_1_2"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E4:60_1_1_1_2"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
;<br/><a NAME="E7:60_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> where <font color="Olive" title="b1">s</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">s</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_1_2"><span class="lab"><font color="Green" title="E42">A8</font></span></a></span>;<br/><a NAME="E8:60_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">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="c4">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>;<br/><a NAME="E9:60_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K15" title="LOPBAN_1:func.15">BoundedLinearOperatorsNorm</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T30" target="_self" title="LOPBAN_1:th.30">Th36</a></span>;<br/><a NAME="E10:60_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A9</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>;<br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span>

<br/>.= 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_1_1_2"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>

;<br/><span class="kw">hence </span><a NAME="E12:60_1_1_1_2"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_1_1_1_2"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:60_1"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:60"/>
 for <font color="Olive" title="b1">t</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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