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<b>let </b><font color="Maroon" title="c2">f</font> be   <a href="hahnban.html#NM1" title="HAHNBAN:NM.1">Functional</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30_6_1"/><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="hahnban.html#V7" title="HAHNBAN:attr.7">0-preserving</a>
 <b>and </b><br/><a NAME="E2:30_6_1"/><i><font color="Green" title="E33">A7</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="hahnban.html#V4" title="HAHNBAN:attr.4">positively_homogeneous</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban.html#D7" target="_self" title="HAHNBAN:def.7">HAHNBAN:def 7</a><br/><b>let </b><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E3:30_6_1"/><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_6_1_1"/>
( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="hahnban.html#E3:30_6_1"><i><font color="Green" title="E34">A8</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_6_1_1_1"><b>suppose </b></a><a NAME="E1:30_6_1_1_1"/><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">x</font></span>)</span> = 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="hahnban.html#E1:30_6_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="hahnban.html#E1:30_6_1_1_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="hahnban.html#D9" target="_self" title="HAHNBAN:def.9">Def9</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_6_1_1_2"><b>suppose </b></a><a NAME="E1:30_6_1_1_2"/>
<font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_6_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban.html#E2:30_6_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="hahnban.html#D6" target="_self" title="HAHNBAN:def.6">Def6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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