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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="rsspace4.html#K4" title="RSSPACE4:func.4">BoundedFunctions</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( 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="rsspace4.html#K4" title="RSSPACE4:func.4">BoundedFunctions</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#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="rsspace4.html#K7" title="RSSPACE4:func.7">PreNorms</a> <span class="p2">(<span class="default"><a href="rsspace4.html#K6" title="RSSPACE4:func.6">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> ) &amp; ( 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="rsspace4.html#K4" title="RSSPACE4:func.4">BoundedFunctions</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#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="rsspace4.html#K7" title="RSSPACE4:func.7">PreNorms</a> <span class="p2">(<span class="default"><a href="rsspace4.html#K6" title="RSSPACE4:func.6">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> ) implies <font color="Maroon" title="c3">NORM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">NORM2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1"/><i><font color="Green" title="E22">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="rsspace4.html#K4" title="RSSPACE4:func.4">BoundedFunctions</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#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="rsspace4.html#K7" title="RSSPACE4:func.7">PreNorms</a> <span class="p2">(<span class="default"><a href="rsspace4.html#K6" title="RSSPACE4:func.6">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="E2:33_1_1"/><i><font color="Green" title="E23">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="rsspace4.html#K4" title="RSSPACE4:func.4">BoundedFunctions</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#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="rsspace4.html#K7" title="RSSPACE4:func.7">PreNorms</a> <span class="p2">(<span class="default"><a href="rsspace4.html#K6" title="RSSPACE4:func.6">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">NORM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">NORM2</font></span><br/>

<a NAME="E3:33_1_1"/><i><font color="Green" title="E24">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">NORM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace4.html#K4" title="RSSPACE4:func.4">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">NORM2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace4.html#K4" title="RSSPACE4:func.4">BoundedFunctions</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="E4:33_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="rsspace4.html#K4" title="RSSPACE4:func.4">BoundedFunctions</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="33_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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K4" title="RSSPACE4:func.4">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> implies <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> )</span><br/>

<b>assume </b><a NAME="E1:33_1_1_1"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K4" title="RSSPACE4:func.4">BoundedFunctions</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></span><br/>

<a NAME="E2:33_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#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1">(<span class="default"><a href="rsspace4.html#K7" title="RSSPACE4:func.7">PreNorms</a> <span class="p2">(<span class="default"><a href="rsspace4.html#K6" title="RSSPACE4:func.6">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="#E1:33_1_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E1:33_1_1_1"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:33_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="#E2:33_1_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E1:33_1_1_1"><i><font color="Green" title="E25">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:33_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="#E3:33_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
