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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> ; <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">Y</font> being  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/> for <font color="Olive" title="b2">f</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="rsspace4.html#K9" title="RSSPACE4:func.9">R_NormSpace_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">Y</font></span>)</span> holds   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="Olive" title="b2">f</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span></span><br/><b>let </b><font color="Maroon" title="c2">Y</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</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">f</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="rsspace4.html#K9" title="RSSPACE4:func.9">R_NormSpace_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> holds   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="Olive" title="b1">f</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span></span><br/><b>let </b><font color="Maroon" title="c3">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="rsspace4.html#K9" title="RSSPACE4:func.9">R_NormSpace_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c3">f</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span></span><br/>





<b>reconsider </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">f</font> as  <a href="rsspace4.html#V1" title="RSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="rsspace4.html#D5" target="_self" title="RSSPACE4:def.5">Def5</a></i>;<br/>
<div><i><font color="Green" title="E27">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K7" title="RSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c4">g</font> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> )</span><br/><b>assume </b><a NAME="E1:39_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K7" title="RSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c4">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font></span><br/><b>consider </b><font color="Maroon" title="c6">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:39_1"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="txt" href="#E1:39_1"><i><font color="Green" title="E28">A2</font></i></a></i>;<br/><b>thus </b><a NAME="E4:39_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="#E3:39_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="normsp_1.html#T8" title="NORMSP_1:th.8">NORMSP_1:8</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>
<a NAME="E3:39"/><i><font color="Green" title="E28">A4</font></i>: 
( not  <a href="rsspace4.html#K7" title="RSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c4">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="rsspace4.html#K7" title="RSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c4">g</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> )
 
<b>by </b><i><a class="ref" href="rsspace4.html#T13" target="_self" title="RSSPACE4:th.13">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">r0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:39"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c5">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace4.html#K7" title="RSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">r0</font> = <font color="Maroon" title="c5">r0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<a NAME="E7:39"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r0</font>
 
<b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="#E5:39"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<a NAME="E8:39"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:39"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E5:39"><i><font color="Green" title="E29">A5</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="E9:39"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c3">f</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="ref" href="rsspace4.html#T17" target="_self" title="RSSPACE4:th.17">Th17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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