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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<a NAME="E1:59"/>
 <a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">c<sub>3</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:59_1"/><i><font color="Green">E40</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:59_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#D9" target="_self">Def9</a></i>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:59_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:59_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> where B is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 </span>}</span> 
 <b>by </b><i><a class="txt" href="lopban_1.html#E1:59_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:59_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><a NAME="E5:59_1_1"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="txt" href="lopban_1.html#E3:59_1"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E6:59_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="txt" href="lopban_1.html#E2:59_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="lopban_1.html#E4:59_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E7:59_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="lopban_1.html#E4:59_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="lopban_1.html#E5:59_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D1">SEQ_4:def 1</a><br/>

<b>thus </b><a NAME="E5:59_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="lopban_1.html#E4:59_1"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:59"/>
( not  <a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="lopban_1.html#K14" target="_self">PreNorms</a> <font color="Maroon">c<sub>3</sub></font> is <a href="seq_4.html#V1">bounded_above</a> )
 ;<br/>


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