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<b>let </b><font color="Maroon" title="c1">GF</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="vectsp_1.html#V12" title="VECTSP_1:attr.12">VectSp-like</a>  <a href="vectsp_1.html#L4" title="VECTSP_1:struct.4">VectSpStr</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>let </b><font color="Maroon" title="c3">V1</font>, <font color="Maroon" title="c4">V2</font>, <font color="Maroon" title="c5">V3</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c3">V1</font> is <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a> &amp; <font color="Maroon" title="c4">V2</font> is <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a> )
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c5">V3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<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">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> : ( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V2</font> ) </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E3:8"/>
for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V3</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V3</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V3</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_4.html#D1" target="_self" title="VECTSP_4:def.1">VECTSP_4:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V3</font>
 <b>and </b><br/><a NAME="E2:8_1"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V3</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">v1</font>, <font color="Maroon" title="c9">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E4:8_1"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">v2</font>
 <b>and </b><br/><a NAME="E5:8_1"/><i><font color="Green" title="E11">A6</font></i>: 
( <font color="Maroon" title="c8">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Maroon" title="c9">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V2</font> )
 <b>by </b><i><a class="txt" href="vectsp_4.html#E2:8"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="vectsp_4.html#E1:8_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">u1</font>, <font color="Maroon" title="c11">u2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E7:8_1"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u2</font>
 <b>and </b><br/><a NAME="E8:8_1"/><i><font color="Green" title="E13">A8</font></i>: 
( <font color="Maroon" title="c10">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Maroon" title="c11">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V2</font> )
 <b>by </b><i><a class="txt" href="vectsp_4.html#E2:8"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="vectsp_4.html#E2:8_1"><i><font color="Green" title="E9">A4</font></i></a></i>;<br/>
<a NAME="E9:8_1"/><i><font color="Green" title="E14">A9</font></i>: 
( <font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Maroon" title="c9">v2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V2</font> )
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="vectsp_4.html#E5:8_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="vectsp_4.html#E8:8_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="vectsp_4.html#D1" target="_self" title="VECTSP_4:def.1">Def1</a></i>;<br/>
<font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">v2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u2</font>
<b>by </b><i><a class="txt" href="vectsp_4.html#E4:8_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="vectsp_4.html#E7:8_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">v2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u1</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="c9">v2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E11:8_1"/>
<font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V3</font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E2:8"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="vectsp_4.html#E9:8_1"><i><font color="Green" title="E14">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font>;<br/>



<b>assume </b><a NAME="E4:8"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V3</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">v1</font>, <font color="Maroon" title="c9">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E6:8"/><i><font color="Green" title="E8">A10</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">v2</font>
 <b>and </b><br/><a NAME="E7:8"/><i><font color="Green" title="E9">A11</font></i>: 
( <font color="Maroon" title="c8">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Maroon" title="c9">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V2</font> )
 <b>by </b><i><a class="txt" href="vectsp_4.html#E2:8"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/>
<a NAME="E8:8"/><i><font color="Green" title="E10">A12</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c8">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c9">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V2</font> )
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="vectsp_4.html#E7:8"><i><font color="Green" title="E9">A11</font></i></a>, <a class="ref" href="vectsp_4.html#D1" target="_self" title="VECTSP_4:def.1">Def1</a></i>;<br/>
<a NAME="E9:8"/>
<font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c8">v1</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="c6">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c9">v2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E6:8"><i><font color="Green" title="E8">A10</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>;<br/>
<b>hence </b><a NAME="E10:8"/>
<font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V3</font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E2:8"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="vectsp_4.html#E8:8"><i><font color="Green" title="E10">A12</font></i></a></i>;<br/>


</div>
