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<div class="add">

<b>let </b><font color="Maroon">F</font> be  <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a> <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_2.html#V5">RightMod-like</a>  <a href="vectsp_2.html#L1">RightModStr</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/>







<a NAME="E1:10"/>
(  not <font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 ;<br/>

<b>assume </b><a NAME="E2:10_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 ;<br/>

<font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_2.html#K1">"</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="vectsp_2.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_2.html#K1">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_2.html#D23">VECTSP_2:def 23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i/>
;<br/>
<a NAME="E4:10_1"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="vectsp_2.html#T43">VECTSP_2:43</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_2.html#D23">VECTSP_2:def 23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:10"/>
( <font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> iff ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> ) )
 <b>by </b><i/>;<br/>


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