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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">V'</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">V'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> #)
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">V'</font>;<br/>

<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="hahnban.html#NM1">Functional</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<a NAME="E3:40"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">f'</font> is <a href="hahnban.html#V2" target="_self">additive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban.html#D4" target="_self">HAHNBAN:def 4</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V'</font> <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:40"/>
<font color="Maroon">f'</font> is <a href="hahnban.html#V3" target="_self">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban.html#D5" target="_self">HAHNBAN:def 5</a><br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V'</font> <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#T4" target="_self">Th4</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:40"/>
<font color="Maroon">f</font> is   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>


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