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<b>let </b><font color="Maroon" title="c3">NORM1</font>, <font color="Maroon" title="c4">NORM2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:59_1_1"/><i><font color="Green" title="E36">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> holds <br/><font color="Maroon" title="c3">NORM1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:59_1_1"/><i><font color="Green" title="E37">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> holds <br/><font color="Maroon" title="c4">NORM2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:59_1_1"/><i><font color="Green" title="E38">A3</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">NORM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">NORM2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:59_1_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> holds <br/><font color="Maroon" title="c3">NORM1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">NORM2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">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" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:59_1_1_1"/><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 ;<br/>

<a NAME="E2:59_1_1_1"/>
<font color="Maroon" title="c3">NORM1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="lopban_1.html#E1:59_1_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="lopban_1.html#E1:59_1_1_1"><i><font color="Green" title="E39">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:59_1_1_1"/>
<font color="Maroon" title="c3">NORM1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">NORM2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="lopban_1.html#E2:59_1_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="lopban_1.html#E1:59_1_1_1"><i><font color="Green" title="E39">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:59_1_1"/>
<font color="Maroon" title="c3">NORM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">NORM2</font>
 <b>by </b><i><a class="txt" href="lopban_1.html#E3:59_1_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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