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<b>let </b><font color="Maroon">NORM1</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="clopban1.html#K8" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>, <a href="numbers.html#K1">REAL</a> , <font color="Maroon">NORM2</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="clopban1.html#K8" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>, <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:59_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K8" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> holds <br/><font color="Maroon">NORM1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="clopban1.html#K12" target="_self">PreNorms</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11" target="_self">modetrans</a> <font color="Olive">x</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:59_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K8" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> holds <br/><font color="Maroon">NORM2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="clopban1.html#K12" target="_self">PreNorms</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11" target="_self">modetrans</a> <font color="Olive">x</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:59_1_1"/><i><font color="Green">E70</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">NORM1</font> <a href="hidden.html#R1">=</a>  <a href="clopban1.html#K8" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">NORM2</font> <a href="hidden.html#R1">=</a>  <a href="clopban1.html#K8" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:59_1_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K8" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> holds <br/><font color="Maroon">NORM1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">NORM2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:59_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K8" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:59_1_1_1"/>
<font color="Maroon">NORM1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="clopban1.html#K12" target="_self">PreNorms</a> <span class="p2">(<span class="default"><a href="clopban1.html#K11" target="_self">modetrans</a> <font color="Maroon">z</font>,<font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="clopban1.html#E1:59_1_1"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E3:59_1_1_1"/>
<font color="Maroon">NORM1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">NORM2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:59_1_1"/>
<font color="Maroon">NORM1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">NORM2</font>
 <b>by </b><i><a class="ref" href="clopban1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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