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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c3">fi</font> be   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c2">X</font>;<br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green" title="E43">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c3">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <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">v</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">q</font> = the <a href="normsp_1.html#U1" title="NORMSP_1:sel.1">norm</a> of <font color="Maroon" title="c1">V</font> as   <a href="hahnban.html#NM2" title="HAHNBAN:NM.2">Banach-Functional</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="hahnban.html#T36" target="_self" title="HAHNBAN:th.36">Th36</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">X</font>;<br/><b>let </b><font color="Maroon" title="c6">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>assume </b><a NAME="E1:44_1"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font>
 ;<br/><a NAME="E2:44_1"/>
<font color="Maroon" title="c4">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</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"><font color="Maroon" title="c6">v</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D1" title="NORMSP_1:def.1">NORMSP_1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:44_1"/>
<font color="Maroon" title="c3">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:44"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="hahnban.html#E1:44_1"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c5">psi</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E5:44"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c5">psi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">fi</font>
 <b>and </b><br/><a NAME="E6:44"/><i><font color="Green" title="E45">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c5">psi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T35" target="_self" title="HAHNBAN:th.35">Th35</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">psi</font>
;<br/>

<b>thus </b><a NAME="E7:44"/>
<font color="Maroon" title="c5">psi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">fi</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E5:44"><i><font color="Green" title="E44">A3</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>

<a NAME="E8:44"/>
<font color="Maroon" title="c4">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</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"><font color="Maroon" title="c6">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D1" title="NORMSP_1:def.1">NORMSP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:44"/>
<font color="Maroon" title="c5">psi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <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="c6">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="txt" href="hahnban.html#E6:44"><i><font color="Green" title="E45">A4</font></i></a></i>;<br/>


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