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<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a><br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="hahnban1.html#K8" target="_self">*'</a> </span>)</span>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="hahnban1.html#K8" target="_self">*'</a> </span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">v</font>, <font color="Maroon">g</font> = <font color="Maroon">w</font> as   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> = 
the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="hahnban1.html#K8" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p2">(<span class="default"><font color="Maroon">V</font> <a href="hahnban1.html#K8" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span>

<br/>.= 
the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="hahnban1.html#K8" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hahnban1.html#K3" target="_self">+</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="hahnban1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hahnban1.html#K3" target="_self">+</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="hahnban1.html#K3" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="hahnban1.html#K8" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="hahnban1.html#K8" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <span class="p2">(<span class="default"><font color="Maroon">V</font> <a href="hahnban1.html#K8" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b><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> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
<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> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="hahnban1.html#K3" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="txt" href="hahnban1.html#E1"><i><font color="Green">Lemma23</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="hahnban1.html#K8" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="hahnban1.html#K8" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <span class="p2">(<span class="default"><font color="Maroon">V</font> <a href="hahnban1.html#K8" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="hahnban1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="hahnban1.html#K8" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i/>
;<br/>


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