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<span class="kw">take </span>
 <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:51_1_1"/>
 for <font color="Olive" title="b1">x</font> being   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font></span>)</span> holds  <font color="Olive" title="b1">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b3">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_2.html#T23" target="_self" title="VECTSP_2:th.23">Th71</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D14" title="VECTSP_1:def.14">VECTSP_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:51_1_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b3">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b3">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_2.html#T23" target="_self" title="VECTSP_2:th.23">Th71</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D15" title="VECTSP_1:def.15">VECTSP_1:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:51_1_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b3">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_2.html#T23" target="_self" title="VECTSP_2:th.23">Th71</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E4:51_1_1"/>
 for <font color="Olive" title="b1">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</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 class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_2.html#T23" target="_self" title="VECTSP_2:th.23">Th71</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D17" title="VECTSP_1:def.17">VECTSP_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E5:51_1_1"/>
(  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="vectsp_2.html#K3" title="VECTSP_2:func.3">LeftModule</a> <font color="Maroon" title="c1">R</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  )
 ; <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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