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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K11" title="C0SP1:func.11">R_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E41">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>

<a NAME="E2:58"/><b>then </b><i><font color="Green" title="E42">A2</font></i>: 
( not  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c3">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c3">f</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> )
 
<b>by </b><i><a class="ref" href="c0sp1.html#T17" target="_self" title="C0SP1:th.17">ThB13</a></i>;<br/>
<a NAME="E3:58"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c3">f</font>
 
;<br/>
<a NAME="E4:58"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="c0sp1.html#K9" title="C0SP1:func.9">PreNorms</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="c0sp1.html#E2:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:58"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">F</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="txt" href="c0sp1.html#E1:58"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="c0sp1.html#T21" target="_self" title="C0SP1:th.21">ThB17</a></i>;<br/>


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