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<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a><br/><b>let </b><font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a> </span>)</span>;<br/>



<b>reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c6">w</font> as   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">w</font></span>)</span> = 
the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="hahnban1.html#K3" title="HAHNBAN1:func.3">+</a> <font color="Maroon" title="c8">g</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="hahnban1.html#K3" title="HAHNBAN1:func.3">+</a> <font color="Maroon" title="c8">g</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hahnban1.html#K3" title="HAHNBAN1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c8">g</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T25" target="_self" title="HAHNBAN1:th.25">Th25</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c8">g</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c8">g</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">w</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hahnban1.html#K3" title="HAHNBAN1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T26" target="_self" title="HAHNBAN1:th.26">Th26</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T27" target="_self" title="HAHNBAN1:th.27">Th27</a></i>
<br/>.= 
the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="ref" href="hahnban1.html#D14" target="_self" title="HAHNBAN1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c5">v</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T28" target="_self" title="HAHNBAN1:th.28">Th28</a></i>
;<br/>


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