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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="rsspace4.html#K9" target="_self">R_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="rsspace4.html#K9" target="_self">R_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E2:41"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">z</font> = <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="funcop_1.html#T58">FUNCOP_1:58</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as  <a href="rsspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E3:41_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">z</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:41_1_1"/>
<font color="Maroon">r</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">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="rsspace4.html#E2"><i><font color="Green">Lemma32</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:41_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">r</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">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 ;<br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="txt" href="rsspace4.html#E3:41_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>
;<br/><b>hence </b><a NAME="E6:41_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="ref" href="rsspace4.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:41_1"/><i><font color="Green">E45</font></i>: 
( not  <a href="rsspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="rsspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font> is <a href="seq_4.html#V1">bounded_above</a> )
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:41_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">r0</font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r0</font> = <font color="Maroon">r0</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="rsspace4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E9:41_1"/><i><font color="Green">E66</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r0</font> &amp; <font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="rsspace4.html#E4:41_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="rsspace4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E10:41_1"/>
( ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="rsspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 ) implies  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="rsspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/>
<a NAME="E11:41_1"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="rsspace4.html#K7" target="_self">PreNorms</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="rsspace4.html#E4:41_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="rsspace4.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="rsspace4.html#D1" target="_self">Def1</a>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E12:41_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rsspace4.html#K8" target="_self">BoundedFunctionsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:41_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">f</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>


</div>
