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<span class="kw">let </span><font color="Maroon" title="c2">NORM1</font>, <font color="Maroon" title="c3">NORM2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="dualsp01.html#K11" title="DUALSP01:func.11">BoundedLinearFunctionals</a> <font color="Maroon" title="c1">X</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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"> ( (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dualsp01.html#K11" title="DUALSP01:func.11">BoundedLinearFunctionals</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">NORM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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="dualsp01.html#K15" title="DUALSP01:func.15">PreNorms</a> <span class="p2">(<span class="default"><a href="dualsp01.html#K14" title="DUALSP01:func.14">Bound2Lipschitz</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dualsp01.html#K11" title="DUALSP01:func.11">BoundedLinearFunctionals</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c3">NORM2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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="dualsp01.html#K15" title="DUALSP01:func.15">PreNorms</a> <span class="p2">(<span class="default"><a href="dualsp01.html#K14" title="DUALSP01:func.14">Bound2Lipschitz</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> ) implies <font color="Maroon" title="c2">NORM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">NORM2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:49_1_2"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dualsp01.html#K11" title="DUALSP01:func.11">BoundedLinearFunctionals</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">NORM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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="dualsp01.html#K15" title="DUALSP01:func.15">PreNorms</a> <span class="p2">(<span class="default"><a href="dualsp01.html#K14" title="DUALSP01:func.14">Bound2Lipschitz</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:49_1_2"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dualsp01.html#K11" title="DUALSP01:func.11">BoundedLinearFunctionals</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c3">NORM2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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="dualsp01.html#K15" title="DUALSP01:func.15">PreNorms</a> <span class="p2">(<span class="default"><a href="dualsp01.html#K14" title="DUALSP01:func.14">Bound2Lipschitz</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</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="c2">NORM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">NORM2</font></span><br/>

<a NAME="E3:49_1_2"/>
 for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dualsp01.html#K11" title="DUALSP01:func.11">BoundedLinearFunctionals</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">NORM1</font> <a href="funct_1.html#K1" title="FUNCT_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="c3">NORM2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">z</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="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dualsp01.html#K11" title="DUALSP01:func.11">BoundedLinearFunctionals</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c2">NORM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">NORM2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:49_1_2_1"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dualsp01.html#K11" title="DUALSP01:func.11">BoundedLinearFunctionals</a> <font color="Maroon" title="c1">X</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="c2">NORM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">NORM2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font></span><br/>

<a NAME="E2:49_1_2_1"/>
<font color="Maroon" title="c2">NORM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</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="dualsp01.html#K15" title="DUALSP01:func.15">PreNorms</a> <span class="p2">(<span class="default"><a href="dualsp01.html#K14" title="DUALSP01:func.14">Bound2Lipschitz</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1_2"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E1:49_1_2_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:49_1_2_1"/>
<font color="Maroon" title="c2">NORM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">NORM2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1_2"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E1:49_1_2_1"><span class="lab"><font color="Green" title="E24">A5</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="E4:49_1_2"/>
<font color="Maroon" title="c2">NORM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">NORM2</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"> verum</span><br/>


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