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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <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> holds <br/> ( <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b2">w</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c6">w</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> )</span><br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">w</font></span>)</span> = 
 the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#T17" target="_self" title="HAHNBAN1:th.17">Th25</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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#U1" title="VECTSP_1:sel.1">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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c6">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> )</span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</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#K3" title="RLVECT_1:func.3">+</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#T18" target="_self" title="HAHNBAN1:th.18">Th26</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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#U1" title="VECTSP_1:sel.1">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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> )</span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</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#K8" title="GROUP_1:func.8">*</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#T19" target="_self" title="HAHNBAN1:th.19">Th27</a></span>
<br/>.= 
 the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D10" target="_self" title="HAHNBAN1:def.10">Def14</a></span>
<br/>.= 
<font color="Maroon" title="c5">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#T20" target="_self" title="HAHNBAN1:th.20">Th28</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:38_1_1"/>
( <font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a>  is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp; <font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a>  is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  &amp; <font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a>  is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp; <font color="Maroon" title="c2">V</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a>  is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D14" title="VECTSP_1:def.14">VECTSP_1:def 14</a>, <a class="ref" href="vectsp_1.html#D15" title="VECTSP_1:def.15">VECTSP_1:def 15</a>, <a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a>, <a class="ref" href="vectsp_1.html#D17" title="VECTSP_1:def.17">VECTSP_1:def 17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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